\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2005-Oujda International Conference on Nonlinear Analysis. \newline {\em Electronic Journal of Differential Equations}, Conference 14, 2006, pp. 207--222.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{207} \begin{document} \title[\hfilneg EJDE/Conf/14 \hfil Fu\v{c}ik spectrum with weights] {Asymmetric elliptic problems in $\mathbb{R}^N$} \author[J.-P. Gossez, L. Leadi \hfil EJDE/Conf/14 \hfilneg] {Jean-Pierre Gossez, Liamidi Leadi} % in alphabetical order \address{Jean-Pierre Gossez \newline D\'epartement de Math\'ematique, C. P. 214, Universit\'e Libre de Bruxelles, 1050 Bruxelles, Belgium} \email{gossez@ulb.ac.be} \address{Liamidi Leadi \newline Institut de Math\'ematiques et de Sciences Physiques, Universit\'e d'Abomey Calavi, 01 BP : 613 Porto-Novo, B\'enin Republic (West Africa)} \email{leadiare@imsp-uac.org, leadiare@yahoo.com} \date{} \thanks{Published September 20, 2006.} \subjclass[2000]{35P30, 35J20, 35J60} \keywords{Principal eigenvalue; indefinite weight; unbounded domain; \hfill\break\indent $p$-Laplacian; Fu\v{c}ik spectrum} \begin{abstract} We work on the whole $\mathbb{R}^N$ and prove the existence of a first nonprincipal eigenvalue for an asymmetric problem with weights involving the $p$-Laplacian (cf. (1.1) below). As an application we obtain a first nontrivial curve in the corresponding Fu\v{c}ik spectrum. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} This work is mainly concerned with the following (asymmetric) eigenvalue problem \begin{equation}\label{b1} -\Delta_{p}u = \lambda [m(x)(u^{+})^{p-1} - n(x)(u^{-})^{p-1}]\quad\text{in }\mathbb{R}^N. \end{equation} Here $\Delta_pu := \mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$, $1p$). A study of (\ref{b1}) together with applications to the Fu\v{c}ik spectrum and to nonresonance was carried out recently in \cite{ari1} in the case of a bounded domain $\Omega\subset\mathbb{R}^N$: \begin{equation}\label{b4} -\Delta_{p}u = \lambda [m(x)(u^{+})^{p-1} - n(x)(u^{-})^{p-1}]\quad\text{in }\Omega,\quad u(x) = 0\quad\text{on }\partial\Omega . \end{equation} Denoting by $\mu_1(m)$ the first positive eigenvalue of the Dirichlet $p$-Laplacian with weight $m$ on $\Omega$ and by $\psi_m$ the associated normalized positive eigenfunction, it was shown in \cite{ari1} that (\ref{b4}) always admits a positive nonprincipal eigenvalue, which in addition is the first eigenvalue of (\ref{b4}) greater than $\mu_1(m)$ and $\mu_1(n)$. This distinguished eigenvalue was constructed by applying a version of the mountain pass theorem to the functional $\int_\Omega |\nabla u|^p$ restricted to the $C^1$ manifold $\{u\in W_0^{1,p}(\Omega): \int_\Omega [m(u^+)^p + n(u^-)^p]=1\}$. In this process the (PS) condition was easily verified by using the $(S)_+$ property of the $p$-Laplacian while the geometry of the mountain pass was derived from the observation that $\psi_m$ and $-\psi_n$ were strict local minima. When trying to adapt the above approach to the case of the whole $\mathbb{R}^N$, the relevant functional is \begin{equation}\label{b5'} J(u):= \int_{\mathbb{R}^N} |\nabla u|^p \end{equation} restricted to \begin{equation}\label{b5} M_{m,n} := \big\{u\in W: B_{m,n}(u) := \int_{\mathbb{R}^N} [m(u^+)^p+ n(u^-)^p] = 1\big\}, \end{equation} where the space $W$ is a suitable weighted Sobolev space on $\mathbb{R}^N$ which will be defined later. One of the main difficulties lies, as expected, in the verification of the (PS) condition. This is carried out in Proposition \ref{prop3.3}, whose proof uses some technique from \cite{all2,hu} as well as a result from \cite{fleck} about the compact imbedding of $W$ into a weighted Lebesgue space. Other difficulties arise in connection with the geometry of the functional (cf. the proof of Proposition \ref{prop3.1}) as well as in the construction of some suitable auxiliary weights (cf. Lemma \ref{lem4.2} and the proof of Theorem \ref{thm4.1}). One should also point out that the study of the continuous dependance of our distinguished eigenvalue of (\ref{b1}) with respect to the weights requires some special care due in particular to the fact that these weights generally do not satisfy any integrability condition on $\mathbb{R}^N$ (cf. Proposition \ref{prop4.7} and Corollary \ref{coro4.8}). The existence of a positive nonprincipal eigenvalue for (\ref{b1}) is derived in Section 3. In Section 4, we prove that the eigenvalue $c(m,n)$ constructed in Section 3 is the first nonprincipal eigenvalue of (\ref{b1}). We also study there some of the properties of $c(m,n)$ as a function of $(m,n)$. Section 5 is devoted to the Fu\v{c}ik spectrum. We show the existence of a first nontrivial curve in $\Sigma\cap (\mathbb{R}^+\times\mathbb{R}^+) $ whose asymptotic behaviour exhibits some similarity with what is happening for the Dirichlet problem on a bounded domain. In the preliminary Section 2, we collect some known results relative to the eigenvalue problem (\ref{b2}) and to various Sobolev imbeddings or Poincar\'e's type inequalities to be used later. \section{Preliminaries} Throughout this work, we write the weights $m$ and $n$ in the form $m=m_1-m_2$, $n=n_1-n_2$, and we assume the following conditions: \begin{itemize} \item[$(H_1)$] $m_1,n_1 \geq 0$, $m_1, n_1 \in L_{\rm loc}^{\infty} (\mathbb{R}^N)\cap L^s(\mathbb{R}^N)$, where $s=N/p$ if $N>p$ and $s=N_0/p$ for some integer $N_0>p$ if $N\leq p$; \item[$(H_2)$] $m_2,n_2 \geq 0$, $m_2, n_2\in L_{\rm loc}^{\infty}(\mathbb{R}^N)$, with in addition $m_2(x), n_2(x)\geq \varepsilon_0$ for some $\varepsilon_0>0$ a.e. in $\mathbb{R}^N$ if $N\leq p$; \item[$(H_3)$] $m^+\not\equiv 0$, $n^+\not\equiv 0$; \item[$(H_4)$] for some $a,b>0$: $am_2(x)\leq n_2(x)\leq bm_2(x)$ a.e. in $\mathbb{R}^N$. \end{itemize} Note that the decomposition $m=m_1-m_2$ does not necessarily coincide with the decomposition $m= m^+ - m^-$. Associated with $m_2$, we define a weighted Sobolev space $W$ as the closure of $C_c^\infty (\mathbb{R}^N)$ with respect to the norm \begin{equation} \|u\|_W: = \Big[\int _{\mathbb{R}^N} (|\nabla u|^p + m_2 |u|^p)\Big]^{1/p}. \end{equation} Note that by $(H_4)$, $n_2$ would lead to the same space $W$. Note also that as observed in \cite{fleck}, the space $W$ does not depend on the decomposition of $m$ into $m_1-m_2$. The following imbeddings hold (cf. e.g. \cite{bre}): $W\hookrightarrow D^{1,p}(\mathbb{R}^N)\hookrightarrow L^{p^*}(\mathbb{R}^N)$ if $N>p$, $W\hookrightarrow W^{1,p}(\mathbb{R}^N)\hookrightarrow L^{q}(\mathbb{R}^N)$ for all $q\in [p, +\infty[$ if $N=p$ and for all $q\in [p,+\infty]$ if $Np$ the closure of $C_c^\infty (\mathbb{R}^N)$ with respect to the norm $(\int_{\mathbb{R}^N}|\nabla u|^p)^{1/p}$ and $p^* :=Np/(N-p)$ is the critical Sobolev exponent. With $s$ as in $(H_1)$ above and $s'$ its H\"{o}lder conjugate, we will denote later by $A$ the constant of the imbedding of $D^{1,p}(\mathbb{R}^N)$ into $L^{ps'}(\mathbb{R}^N)=L^{p^*}(\mathbb{R}^N)$ when $N>p$, and by $B$ the constant of the imbedding of $W^{1,p}(\mathbb{R}^N)$ into $L^{ps'}(\mathbb{R}^N)$ when $N\leq p$. One also has the compact imbedding of $W$ into $L^p(m_1 ,\mathbb{R}^N)$, the $L^p$ space on $\mathbb{R}^N$ with weight $m_1$ (cf. \cite{fleck}). By a solution $u$ of (\ref{b1}) (or of related equations), we mean a weak solution, i.e. $u\in W$ with \begin{equation}\label{b6} \int_{\mathbb{R}^N}|\nabla u|^{p-2}\nabla u \nabla v = \lambda\int_{\mathbb{R}^N}[m(u^+)^{p-1} - n(u^-)^{p-1}]v\quad \forall v\in W . \end{equation} Note that by the above imbeddings, every integral in (\ref{b6}) is well-defined. Regularity results from \cite{ser} and \cite{tolk} on general quasilinear equations imply that such a weak solution $u$ belongs to $C^1(\mathbb{R}^N)$. It is also known that if $Np$ and $(H_4)$ is replaced by $(H_4')$ (cf. Remark \ref{rmk2.5} below), then a weak solution $u$ decays to zero at infinity (cf. \cite{bre} for $Np$). Let us define $$ \lambda_1(m):=\inf\big\{\int_{\mathbb{R}^N}|\nabla u|^p: u\in W \text{ and } \int_{\mathbb{R}^N}m|u|^p = 1\big\}. $$ It is known (cf. \cite{all2, dra1,dra2,fleck, fleck1}) that this infimum is achieved and that $\lambda_1(m)$ is the unique positive principal eigenvalue of (\ref{b2}). (By a principal eigenvalue, we mean an eigenvalue associated to an eigenfunction which does not change sign). Moreover $\lambda_1(m)$ is simple and admits an eigenfunction $\varphi_m\in W\cap C^1(\mathbb{R}^N)$, with $\varphi_m(x)>0$ in $\mathbb{R}^N$ and $\int_{\mathbb{R}^N}m\varphi_m^p = 1$. One also knows that $\lambda_1(m)$ is isolated in the spectrum, which implies \begin{equation}\label{b6'} \lambda_2(m):=\inf\left\{\lambda\in \mathbb{R}:\lambda \text{ eigenvalue of (\ref{b2}) with } \lambda>\lambda_1(m)\right\}>\lambda_1(m). \end{equation} As we will see later (cf. Remark \ref{rmk3.4} or Theorem \ref{thm4.1}), this infimum (\ref{b6'}) is also achieved, and consequently $\lambda_2(m)$ is really the second positive eigenvalue of (\ref{b2}). \begin{remark} \label{rmk2.1}\rm Assume that $m$ satisfies $(H_1)$, $(H_2)$, $(H_3)$ and that $m^-\not\equiv 0$. If $N>p$ and $m_2\in L^{N/p}(\mathbb{R}^N)$, then (\ref{b2}) also has a unique negative principal eigenvalue $\lambda_{-1}(m)= -\lambda_1(-m)$. If $N\leq p$, then (\ref{b2}) generally does not admit a negative principal eigenvalue; this follows from the nonexistence results of \cite{bro} for $p=2$ and \cite{hu} for $p\not=2$. \end{remark} The following lemma will play a role in low dimensions. It can be easily derived from the proof in \cite[Theorem 3]{all2}. (The assumptions about $N_0$ in $(H_1)$ and about $\varepsilon_0$ in $(H_2)$ are used here). \begin{lemma} \label{lem2.2} Let $N\leq p$. There exists $C=C(m_1,m_2,n_1,n_2,N,p,N_0)$ such that \begin{equation}\label{b7} \int_{\mathbb{R}^N}|u|^p\leq C\int_{\mathbb{R}^N}|\nabla u|^p \end{equation} for all $u\in W$ satisfying $B_{m,n}(u)\geq 0$. Moreover the constant $C$ in (\ref{b7}) can be chosen so as to remain bounded when $m_1$ and $n_1$ vary in a bounded subset of $L^{N_0/p}(\mathbb{R}^N)$. \end{lemma} \begin{remark} \label{rmk2.3} \rm It suffices in all this work to assume that the inequalities in $(H_2)$ and $(H_4)$ hold ``at infinity". More precisely denote by $(H_2)_R$ and $(H_4)_R$ the same conditions as $(H_2)$ and $(H_4)$ except that the inequalities $m_2(x), n_2(x)\geq \text{some } \varepsilon_0$ and $am_2(x)\leq n_2(x)\leq bn_2(x)$ are assumed to hold only for a.e. $x$ with $|x|\geq R$ for some $R\geq 0$. Suppose now that $m=m_1-m_2$, $n=n_1-n_2$ satisfy $(H_1)$, $(H_2)_R$, $(H_3)$, $(H_4)_R$. By writing $m=\tilde{m}_1-\tilde{m}_2$, $n=\tilde{n}_1-\tilde{n}_2$ where $\tilde{m}_1$, $\tilde{m}_2$, $\tilde{n}_1$, $\tilde{n}_2$ are obtained from $m_1$, $m_2$, $n_1$, $n_2$ by adding $1_{B_R}$ (the characteristic function of the ball with center $0$ and radius $R$), one easily sees that $m=\tilde{m}_1-\tilde{m}_2$, $n=\tilde{n}_1-\tilde{n}_2$ now satisfy $(H_1)$, $(H_2)$, $(H_3)$, $(H_4)$. \end{remark} \begin{remark} \label{rm2.4} \rm In the situation of Remark \ref{rmk2.3}, one can also show that the space $W$ associated to $m_2$ coincides with the space $W$ associated to $\tilde{m}_2$. (The proof of this fact uses the inequality that if $\Omega$ is a smooth bounded domain and if $E$ is a subset of $\Omega$ of positive measure, then there exists a constant $c$ such that $\|u\|_{L^p(\Omega)}\leq c (\|u\|_{L^p(E)} + \|\nabla u\|_{L^p(\Omega)})$ for all $u\in W^{1,p}(\Omega)$). It follows from this observation that the space $W$ does not depend on the decomposition of $m$ into $m_1-m_2$ when $m_1,m_2$ satisfy $(H_1)$ and $(H_2)_R$. \end{remark} \begin{remark} \label{rmk2.5} \rm In high dimensions, assumption $(H_4)$ can be replaced in all this work by \begin{itemize} \item[$(H_4')$] $N>p$ and $ m_2, n_2\in L^{N/p}(\mathbb{R}^N)$. \end{itemize} This situation is in fact much simpler (for instance $W$ is then equal to $D^{1,p}(\mathbb{R}^N)$). Without $(H_4)$, $(H_4)_R$ (i.e. when $m_2$ and $n_2$ are unrelated) or $(H_4')$, it is not clear how to deal with the asymmetric problems (\ref{b1}) and (\ref{b3}). \end{remark} Let us conclude this section with some general definitions relative to the (PS) condition. Let $E$ be a real Banach space and \begin{equation}\label{b8} M:=\{u\in E:g(u)=1\}, \end{equation} where $g\in C^1(E, \mathbb{R})$ and $1$ is a regular value of $g$. Let $f\in C^1(E, \mathbb{R})$ and denote by $\tilde{f}$ the restriction of $f$ to $M$. The differential of $\tilde{f}$ at $u\in M$ has a norm which will be denoted by $\|\tilde{f}'(u)\|_*$ and which is given by the norm of the restriction of $f'(u)\in E^*$ to the tangent space of $M$ at $u$: $$ T_uM:=\{v\in E:\langle g'(u),v\rangle =0\}, $$ where $\langle ,\rangle$ denotes the pairing between $E$ and its dual $E^*$. A critical point of $\tilde{f}$ is a point $u\in M$ such that $\|\tilde{f}'(u)\|_* =0$; $\tilde{f}(u)$ is then called a critical value of $\tilde{f}$. We recall that $\tilde{f}$ is said to satisfy the (PS) condition if for any sequence $u_k\in M$ such that $\tilde{f}(u_k)$ is bounded and $\|\tilde{f}'(u_k)\|_*\to 0$, one has that $u_k$ admits a convergent subsequence. \section{Construction of a nonprincipal eigenvalue} In this section and in the following one, we consider the eigenvalue problem (\ref{b1}). It will always be assumed that the weights $m$ and $n$ satisfy the hypothesis $(H_1)$, $(H_2)$, $(H_3)$ and $(H_4)$. We look for eigenvalues $\lambda$ of (\ref{b1}) with $\lambda>0$. Clearly the only positive principal eigenvalues of (\ref{b1}) are $\lambda_1(m)$ and $\lambda_1(n)$. Moreover multiplying by $u^+$ or $u^-$, one easily sees that if (\ref{b1}) with $\lambda>0$ has a solution which changes sign, then $\lambda>\max\{\lambda_1(m) , \lambda_1(n)\}$. Proving the existence of such a solution which changes sign is our purpose in this section. We will use a variational approach and consider the functionals $J$ and $B_{m,n}$ defined in (\ref{b5'}) and (\ref{b5}), which are $C^1$ functionals on $W$, and the restriction $\tilde{J}$ of $J$ to the manifold $M_{m,n}$ defined in (\ref{b5}). In this context one easily verifies that $\lambda>0$ is an eigenvalue of (\ref{b1}) if and only if $\lambda$ is a critical value of $\tilde{J}$. A first critical point of $\tilde{J}$ comes from global minimization. Indeed $$ \tilde{J}(u)\geq \lambda_1(m)\Big[\int_{\mathbb{R}^N}m(u^+)^p\Big]^+ + \lambda_1(n)\Big[\int_{\mathbb{R}^N}n(u^-)^p\Big]^+ \geq\min\{\lambda_1(m) , \lambda_1(n)\} $$ for all $u\in M_{m,n}$, and one has $\tilde{J}(u) = \min\{\lambda_1(m) , \lambda_1(n)\}$ for either $u=\varphi_m$ or $u=-\varphi_n$. Consequently either $\varphi_m$ or $-\varphi_n$ is a global minimum of $\tilde{J}$ and so a critical point of $\tilde{J}$. A second critical point of $\tilde{J}$ comes from the following proposition. \begin{proposition} \label{prop3.1} $\varphi_m$ and $-\varphi_n$ are strict local minimum of $\tilde{J}$, with corresponding critical values $\lambda_1(m)$ and $\lambda_1(n)$. \end{proposition} \begin{proof} The present proof is partly different from that of the analogous result in \cite{ari1}; the difficulty lies at the level of \cite[Lemma 3]{ari1}. We adapt to our situation some technique from \cite{dra1}. Let us consider $\varphi_m$ (similar argument for $-\varphi_n$). Assume by contradiction the existence of a sequence $u_k\in M_{m,n}$ with $u_k\not= \varphi_m$, $u_k\to \varphi_m$ in $W$ and $\tilde{J}(u_k)\leq \lambda_1(m)$. We first observe that $u_k$ changes sign for $k$ sufficiently large. Indeed, since $u_k\to\varphi_m$, $u_k$ must be $>0$ somewhere. If $u_k\geq 0$ in $\mathbb{R}^N$, then $$ \tilde{J}(u_k)= \int_{\mathbb{R}^N}|\nabla u_k|^p > \lambda_1(m)\int_{\mathbb{R}^N}mu_k^p = \lambda_1(m) $$ since $u_k\not= \varphi_m$ and $u_k\in M_{m,n}$. But this contradicts $\tilde{J}(u_k)\leq \lambda_1(m)$. So $u_k$ changes sign for $k$ sufficiently large. Now we have \begin{align*} \lambda_1(m)\int_{\mathbb{R}^N}[m(u_k^+)^p+ n(u_k^-)^p] &=\lambda_1(m) \geq\tilde{J}(u_k) \\ &=\int_{\mathbb{R}^N}|\nabla u_k^+|^p + \int_{\mathbb{R}^N}|\nabla u_k^-|^p\\ &\geq \lambda_1(m)\int_{\mathbb{R}^N}m(u_k^+)^p + \int_{\mathbb{R}^N}|\nabla u_k^-|^p. \end{align*} Consequently \begin{equation}\label{b9} \lambda_1(m)\int_{\mathbb{R}^N} n(u_k^-)^p\geq \int_{\mathbb{R}^N}|\nabla u_k^-|^p . \end{equation} Let $v_k:=u_k^-/(\int_{\mathbb{R}^N}|\nabla u_k^-|^p)^{1/p}$ and $\Omega_k^-:=\{x\in \mathbb{R}^N:u_k(x)<0\}$. We deduce from (\ref{b9}) that \begin{equation}\label{b10} \frac{1}{\lambda_1(m)}\leq \int_{\mathbb{R}^N}n(v_k)^p\leq \int_{\Omega_k^-} n_1(v_k)^p. \end{equation} Consider first the case $N>p$. We deduce from (\ref{b10}), using H\"{o}lder inequality, that $$\frac{1}{\lambda_1(m)}\leq \|n_1\|_{L^s(\Omega_k^-)}\|v_k\|_{L^{p^*}(\mathbb{R}^N)}^p\leq A^p \|n_1\|_{L^s(\Omega_k^-)},$$ where the imbedding constant $A$ was defined in Section 2, and consequently \begin{equation}\label{b11} \|n_1\|_{L^s(\Omega_k^-)}\geq \frac{1}{A^p\lambda_1(m)}=\varepsilon. \end{equation} Take now $r>0$ sufficiently large so that $ \|n_1\|_{L^s(B_r^c)}^s\leq \varepsilon^s/2$, where $B_r^c = \mathbb{R}^N\backslash B_r$ and $B_r$ denotes the ball of radius $r$ centred at the origin. We deduce from (\ref{b11}) that $ \|n_1\|_{L^s(\Omega_k^-\cap B_r)}^s\geq \varepsilon^s/2$, and consequently \begin{equation}\label{b12} |\Omega_k^-\cap B_r|\geq \frac{\varepsilon^s}{2\|n_1\|_{L^\infty (B_r)}^s}>0, \end{equation} where $|E|$ denotes the measure of the set $E$. Since $u_k\to \varphi_m$ in $L^p(B_r)$ and $\varphi_m(x)>0$ for all $x\in B_r$, one has that $|\{x\in B_r:u_k(x)<0\}|\to 0$. But this contradicts (\ref{b12}). In the case $N\leq p$, we have a similar situation. Indeed using H\"{o}lder inequality, the imbedding of $W^{1,p}(\mathbb{R}^N)$ into $L^{ps'}(\mathbb{R}^N)$ (with constant $B$, cf. Section 2) and Lemma \ref{lem2.2} (with constant $C$), one derives from (\ref{b10}) that \begin{equation}\label{b12'} \frac{1}{\lambda_1(m)}\leq (1+C)B^p\|n_1\|_{L^s(\Omega_k^-)}. \end{equation} The conclusion then follows as in the case $N>p$. \end{proof} To get a third critical point of $\tilde{J}$, we will use a version of the mountain pass theorem on a $C^1$ manifold. Let us introduce the following family of paths in the manifold $M_{m,n}$: \begin{equation}\label{b13} \Gamma:=\{\gamma\in C([-1,1], M_{m,n}):\gamma(-1)= \varphi_m \text{ and } \gamma(1)=-\varphi_n\}. \end{equation} Arguing as in \cite[p. 589 ]{ari1}, one shows that $\Gamma$ is nonempty, and so the minimax value \begin{equation}\label{b14} c(m,n):=\inf_{\gamma\in\Gamma}\max_{u\in\gamma([-1,1])}\tilde{J}(u), \end{equation} is finite. The following is the main result in this section. \begin{theorem} \label{thm3.2} $c(m,n)$ is an eigenvalue of \eqref{b1} which satisfies \begin{equation}\label{b15} \max\{\lambda_1(m) , \lambda_1(n)\}< c(m,n). \end{equation} \end{theorem} The rest of this section is devoted to the proof of the above theorem. We first consider the (PS) condition. \begin{proposition} \label{prop3.3} The functional $\tilde{J}$ satisfies the (PS) condition on $M_{m,n}$ \end{proposition} \begin{proof} Let $u_k\in M_{m,n}$ be a (PS) sequence for $\tilde{J}$. So $\int_{\mathbb{R}^N}|\nabla u_k|^p$ remains bounded and for some $\varepsilon_k\to 0$, \begin{equation}\label{b16} \big|\int_{\mathbb{R}^N}|\nabla u_k|^{p-2}\nabla u_k\nabla w\big|\leq \varepsilon_k\|w\|_W \end{equation} for all $w\in T_{u_k}(M_{m,n})$. We will first prove that $u_k$ remains bounded in $W$. In case $N>p$, $u_k$ clearly remains bounded in $D^{1,p}(\mathbb{R}^N)$ and consequently in $L^{p^*}(\mathbb{R}^N)$. Using $B_{m,n}(u_k)=1$ and $(H_4)$, one has \begin{equation}\label{b17'} \min(a, 1)\int_{\mathbb{R}^N}m_2|u_k|^p\leq -1 + \int_{\mathbb{R}^N}[m_1(u_k^+)^p + n_1(u_k^-)^p], \end{equation} where the right hand side remains bounded (by $(H_1)$ and H\"{o}lder inequality). Consequently $\int_{\mathbb{R}^N}m_2|u_k|^p$ remains bounded. In the case $N\leq p$, Lemma \ref{lem2.2} implies that $u_k$ remains bounded in $W^{1,p}(\mathbb{R}^N)$ and consequently in $L^{ps'}(\mathbb{R}^N)$. One then again deduces from (\ref{b17'}) that $\int_{\mathbb{R}^N}m_2|u_k|^p$ remains bounded. Hence in any case $N>p$ or $N\leq p$, $u_k$ remains bounded in $W$. It follows that for a subsequence (still denoted by $u_k$), $u_k\to u$ weakly in $W$, strongly in $L^p(m_1, \mathbb{R}^N)$ and in $L^p(n_1, \mathbb{R}^N)$, and $\int_{\mathbb{R}^N}|\nabla u_k|^p$ converges. In the rest of the proof we will assume $N>p$ (a similar argument holds in the case $N\leq p$). Observe that if $w\in W$, then $(w-a_k(w)u_k)\in T_{u_k}(M_{m,n})$ where $a_k(w):=\int_{\mathbb{R}^N}[m(u_k^+)^{p-1} - n(u_k^-)^{p-1}]w$. Putting $w= (u_k-u_l)- a_k(u_k-u_l)u_k$ in (\ref{b16}), one deduces $$ \int_{\mathbb{R}^N}|\nabla u_k|^{p-2}\nabla u_k\nabla (u_k-u_l) = t_k\int_{\mathbb{R}^N}[m(u_k^+)^{p-1} - n(u_k^-)^{p-1}](u_k-u_l) + 0(\varepsilon_k), $$ where $t_k:=\int_{\mathbb{R}^N}|\nabla u_k|^p$. This implies \begin{equation} \label{b19} \begin{aligned} 0&\leq \int_{\mathbb{R}^N}(|\nabla u_k|^{p-2}\nabla u_k - |\nabla u_l|^{p-2}\nabla u_l)\nabla (u_k-u_l)\\ &= t_k\int_{\mathbb{R}^N} m[(u_k^+)^{p-1}-(u_l^+)^{p-1}](u_k-u_l) \\ &\quad +t_k\int_{\mathbb{R}^N} n[-(u_k^-)^{p-1}+(u_l^-)^{p-1}](u_k-u_l)\\ &\quad +(t_k-t_l)\int_{\mathbb{R}^N} [m(u_l^+)^{p-1}-n(u_l^-)^{p-1}](u_k-u_l) + 0(\varepsilon_k)+ 0(\varepsilon_l)\\ &\leq t_k(I_1 + I_2) + |t_k-t_l|I_3 +0(\varepsilon_k)+ 0(\varepsilon_l), \end{aligned} \end{equation} where \begin{gather*} I_1:=\int_{\mathbb{R}^N} m_1[(u_k^+)^{p-1}-(u_l^+)^{p-1}](u_k-u_l), \\ I_2:=\int_{\mathbb{R}^N} n_1[-(u_k^-)^{p-1}+(u_l^-)^{p-1}](u_k-u_l) ,\\ I_3:=\int_{\mathbb{R}^N}|m(u_l^+)^{p-1}-n(u_l^-)^{p-1}\|u_k-u_l|. \end{gather*} We claim that the right hand side of (\ref{b19}) approaches zero when $k,l\to +\infty$. Indeed using H\"{o}lder inequality and the strong convergence of $u_k$ in $L^p(m_1, \mathbb{R}^N)$, one sees that $I_1\to 0$. Similarly $I_2\to 0$ . Furthermore H\"{o}lder inequality implies that $I_3$ remains bounded. Since $(t_k-t_l)\to 0$, we conclude that the right hand side of (\ref{b19}) goes to $0$ as $k,l\to +\infty$, and the claim is proved . To go on in the proof of Proposition \ref{prop3.3}, we observe that for some constant $d=d(p)$ and for any $\alpha, \beta\in \mathbb{R}^N$, \begin{equation}\label{b21} |\alpha-\beta|^p\leq d\{(|\alpha|^{p-2}\alpha-|\beta|^{p-2}\beta)(\alpha-\beta)\}^{r/2}(|\alpha|^p +|\beta|^p)^{1-r/2}, \end{equation} where $r=p$ if $p\in ]1,2[$ and $r=2$ if $p\geq 2$ (cf. \cite{lin}). Applying (\ref{b21}) and H\"{o}lder inequality, one easily derives from the claim that $\nabla u_k \to \nabla u$ in $L^p(\mathbb{R}^N)$. Moreover the calculation leading to (\ref{b19}) gives, using $(H_4)$, \begin{eqnarray*} 0&\leq& \min (a, 1) t_k\int_{\mathbb{R}^N}m_2(|u_k|^{p-2}u_k - |u_l|^{p-2}u_l)(u_k-u_l)\\ &\leq& t_k(I_1 +I_2)+ |t_k-t_l|I_3 + 0(\varepsilon_k)+0(\varepsilon_l) , \end{eqnarray*} where the right hand side goes to zero by the claim. We then deduce from the above that $\int_{\mathbb{R}^N} m_2|u_k- u_l|^p\to 0$ by applying successively (\ref{b21}), H\"{o}lder inequality and the fact that $\lim_{k\to +\infty}t_k= \int_{\mathbb{R}^N}|\nabla u|^p\not= 0$ (the latter quantity is nonzero because $u\in W$ and $W$ does not contain any nonzero constant). Consequently $u_k\to u$ in $W$ and Proposition \ref{prop3.3} is proved. \end{proof} \begin{remark} \label{rmk3.4} \rm The arguments in the above proof can be used to show that the positive part of the spectrum of (\ref{b2}) is closed (cf. chap.2 of \cite{lea} for details). \end{remark} We now have all the ingredients for the next proof. \begin{proof}[Proof of Theorem \ref{thm3.2}] The conclusion follows by applying the mountain pass theorem on a $C^1$ manifold as given in \cite[Proposition 4]{ari1} or in \cite[Proposition 2.1]{mab2}: the (PS) condition is provided by Proposition \ref{prop3.3} and the geometry comes by combining Proposition \ref{prop3.1} with \cite[Lemma 6]{ari1}. \end{proof} \section{A first nonprincipal eigenvalue} We have seen at the beginning of Section 3 that $\lambda_1(m)$ and $\lambda_1(n)$ are the first two positive eigenvalues of (\ref{b1}). The present section is mainly devoted to the proof that the eigenvalue $c(m,n)$ constructed in \eqref{b14} is the next positive eigenvalue of (\ref{b1}). \begin{theorem} \label{thm4.1} Problem (\ref{b1}) does not admit any eigenvalue between the values $\max\{\lambda_1(m),\lambda_1(n)\}$ and $c(m,n)$. \end{theorem} \begin{proof} The present proof is partly different from that of the analogous result in \cite{ari1}; the difficulty lies at the level of the construction of some auxiliary weights. Assume by contradiction that there exists an eigenvalue $\lambda$ of problem (\ref{b1}) with $\max\{\lambda_1(m),\lambda_1(n)\}<\lambda < c(m,n)$. Our goal is to construct a path in $\Gamma$ on which $\tilde{J}$ remains $\leq \lambda$, which yields a contradiction with the definition \eqref{b14} of $c(m,n)$. Let $u\in M_{m,n}$ be a critical point of $\tilde{J}$ at level $\lambda$. Since $u$ changes sign, one obtains from the equation satisfied by $u$, \begin{equation}\label{b23} 0<\int_{\mathbb{R}^N}|\nabla u^+|^p= \lambda\int_{\mathbb{R}^N} m(u^+)^p \text{ and } 0<\int_{\mathbb{R}^N}|\nabla u^-|^p= \lambda\int_{\mathbb{R}^N}n(u^-)^p. \end{equation} The desired path will be constructed in several steps, using $u$ as starting point. First we go from $u$ to $v:=u^+/B_{m,n}(u^+)^{1/p}\equiv (u^+)_{m,n}$ by writing \begin{equation} \gamma_1(t) := \left[tu + (1-t)u^+ \right]_{m,n}, t\in [0,1]. \end{equation} Using (\ref{b23}), it is easy to show that $\gamma_1(t)$ is well-defined, belongs to $M_{m,n}$ and satisfies $\tilde{J}(\gamma_1(t)) = \lambda ~~\forall t\in [0,1]$. In a similar way we go from $u$ to $(-u^-)_{m,n}$ in $M_{m,n}$ by staying at level $\lambda$. We now describe the construction of a path in $M_{m,n}$ from $v$ to $\varphi_m$ which stays at levels $\leq \lambda$. A similar construction would yield a path in $M_{m,n}$ from $(-u^-)_{m,n}$ to $-\varphi_n$ which stays at levels $\leq \lambda$. Putting everything together, we get the desired path from $\varphi_m$ to $-\varphi_n$. To construct the path from $v$ to $\varphi_m$, first consider the manifold $M_{m,m}$. Clearly $v\in M_{m,m}$. The critical points of the restriction of $J$ to $M_{m,m}$ are the normalized eigenfunctions of (\ref{b2}). Since $v$ does not change sign and vanishes on a set of positive measure, $v$ is not a critical point of this restriction of $J$ to $M_{m,m}$. Consequently there exists a $C^1$ path $\nu : ]-\epsilon, \epsilon[ \to M_{m,m}$ with $\nu(0)=v$ and $\frac{d}{dt}J(\nu(t))\big|_{t=0}\not= 0$. Following a little portion of this path $\nu$ in the positive or negative direction ( call $\nu_1$ that portion), we move from $v$ to a point $w$ by a path in $M_{m,m}$ which, with the exception on its starting point $v$ where $J(v)= \lambda$, lies at levels $<\lambda$. The path $\gamma_2(t)= |\nu_1(t)|$ then lies in $M_{m,n}$ (because it lies in $M_{m,m}$ and is made of nonnegative functions), goes from $v$ to $v_1:= |w|$ and remains, with the exception of its starting point $v$ where $J(v)= \lambda$, at levels $<\lambda$ (since $J(|\nu_1(t)|)= J(\nu_1(t))$ ). Let now $m_{(\epsilon)}$ be defined for $0\leq \epsilon\leq 1$ by \begin{equation*} m_{(\epsilon)} = \begin{cases} m& \text{if } m<0\\ \epsilon m_1-m_2 & \text{if } m\geq 0. \end{cases} \end{equation*} One has $m_{(1)}\equiv m$, $m_{(1)}^+\not\equiv 0$, $m_{(0)}\leq 0$ and so $m_{(0)}^+\equiv 0$. Hence there exists $\epsilon_0\in ]0,1[$ such that \begin{gather*} m_{(\epsilon)}^+\not\equiv 0 \quad \text{if } \epsilon_0<\epsilon<1, \\ m_{(\epsilon)}^+\equiv 0 \quad \text{if } 0\leq \epsilon\leq\epsilon_0 . \end{gather*} Using Lemma \ref{lem4.2} below, we see that for $\epsilon>\epsilon_0$ close to $\epsilon_0$, $m_{(\epsilon)}$ will be a weight $l$ of the form $l_1-l_2$ such that $m,l$ satisfy $(H_1)$, $(H_2)$, $(H_3)$ and $(H_4)$, with in addition $\lambda_1(l)>\lambda$ and $l\leq m$ in $\mathbb{R}^N$. Fix such an $\epsilon$. We then consider the manifold $M_{m,l}$ and the sublevel set $$ \mathcal{O}:=\{u\in M_{m,l}:J(u)<\lambda\} . $$ Clearly $v_1$ and $\varphi_m\in \mathcal{O}$ (because they belong to $M_{m,m}$, are $\geq 0$ and have the right levels). Moreover the only critical point in $\mathcal{O}$ of the restriction of $J$ to $M_{m,l}$ is $\varphi_m$ (because the first two critical levels $\lambda_1(m)$ and $\lambda_1(l)$ verify $\lambda_1(m)<\lambda<\lambda_1(l)$). Applying \cite[Lemma 14]{ari1} to the component of $\mathcal{O}$ which contains $v_1$, we get a path $\gamma_3$ in $\mathcal{O}$ from $v_1$ to $\varphi_m$. We then consider the path $$ \gamma_4(t):= \frac{|\gamma_3(t)|}{(\int_{\mathbb{R}^N}m |\gamma_3(t)|^p)^{1/p}} . $$ By the choice of $l$, one has $$ 1=\int_{\mathbb{R}^N} [m(\gamma_3(t)^+)^p +l(\gamma_3(t)^-)^p]\leq \int_{\mathbb{R}^N} [m(\gamma_3(t)^+)^p +m(\gamma_3(t)^-)^p] = \int_{\mathbb{R}^N}m|\gamma_3(t)|^p, $$ and consequently $\gamma_4$ is well-defined. Moreover $\gamma_4$ goes from $v_1$ to $\varphi_m$ and belongs to $M_{m,n}$. Finally $$ J(\gamma_4(t))= \frac{\int_{\mathbb{R}^N}|\nabla\gamma_3(t)|^p} {\int_{\mathbb{R}^N}m|\gamma_3(t)|^p} \leq \int_{\mathbb{R}^N}|\nabla\gamma_3(t)|^p < \lambda, $$ since $\gamma_3(t)\in \mathcal{O}$. The path $\gamma_4$ thus allows us to move from $v_1$ to $\varphi_m$ in $M_{m,n}$ by staying at levels $<\lambda$. \end{proof} \begin{lemma} \label{lem4.2} $\lambda_1[m_{(\epsilon)}]\to \infty$ as $\epsilon\downarrow\epsilon_0$. \end{lemma} \begin{proof} One has \begin{equation}\label{b24} \frac{1}{\lambda_1[m_{(\epsilon)}]} =\frac{\int_{\mathbb{R}^N}m_{(\epsilon)}\varphi_{m_{(\epsilon)}}^p }{\int_{\mathbb{R}^N}|\nabla \varphi_{m_{(\epsilon)}}|^p} \leq \frac{\|(\epsilon m_1-m_2)^+\|_{L^s(\mathbb{R}^N)} \|\varphi_{m_{(\epsilon)}}\|_{L^{ps'}(\mathbb{R}^N)}^p}{\int_{\mathbb{R}^N}|\nabla \varphi_{m_{(\epsilon)}}|^p}. \end{equation} If $N>p$, one deduces from (\ref{b24}) that $ 1/\lambda_1[m_{(\epsilon)}]\leq A \|(\epsilon m_1-m_2)^+\|_{L^s(\mathbb{R}^N)}$, and the conclusion follows since $(\epsilon m_1-m_2)^+\in L^s(\mathbb{R}^N)$ and $(\epsilon m_1-m_2)^+\downarrow 0$ as $\epsilon\downarrow\epsilon_0$. If $N\leq p$, since $B_{m_{(\epsilon)},n_{(\epsilon)}}(\varphi_{m_{(\epsilon)}})= 1$, one has $$ \|\varphi_{m_{(\epsilon)}}\|_{L^{ps'}(\mathbb{R}^N)}^p\leq B^p(1+ C(\epsilon))\int_{\mathbb{R}^N} |\nabla \varphi_{m_{(\epsilon)}}|^p, $$ where $C(\epsilon) = C(m_{(\epsilon)},N,p,N_0)$ is the constant from Lemma \ref{lem2.2} (which remains bounded as $\epsilon$ varies in $]\epsilon_0, 1[$) . The conclusion then follows from (\ref{b24}) as in the case $N>p$. \end{proof} Theorem \ref{thm4.1} for $m\equiv n$ yields the following variational characterization of the second eigenvalue of the $p$-Laplacian with weight on $\mathbb{R}^N$ (cf. (\ref{b6'})). \begin{corollary} \label{coro4.3} One has $$ \lambda_2(m)= \inf_{\gamma\in \Gamma_0}\max_{u\in \gamma([-1,1])}\int_{\mathbb{R}^N}|\nabla u|^p, $$ where $\Gamma_0:=\{\gamma\in C([-1,1], M_{m,m}):\gamma(-1)= \varphi_m \text{ and } \gamma(1)=-\varphi_m\}$ and $ M_{m,m}:=\{u\in W:\int_{\mathbb{R}^N}m|u|^p=1\}$. \end{corollary} We conclude this section with some properties of the eigenvalue $c(m,n)$ as a function of the weights $m,n$. The following slightly different variational characterization of $c(m,n)$ will be useful for this purpose. It can be obtained by an easy adaptation of arguments in \cite{ari1}. \begin{proposition} \label{prop4.4} One has \begin{equation}\label{b24'} c(m,n) = \inf_{\gamma\in \Gamma_1}\max_{t\in [-1,1]}J(\gamma(t)), \end{equation} where $ \Gamma_1 := \{\gamma\in C([-1,1], M_{m,n}):\gamma(-1)\geq 0 \text{ and } \gamma(1)\leq 0\}$. \end{proposition} \begin{proposition} \label{prop4.5} Let $m=m_1 -m_2$, $n=n_1-n_2$, $\hat{m}= \hat{m}_1-\hat{m}_2$, $\hat{n}= \hat{n}_1-\hat{n}_2$, and assume that hypothesis $(H_1)$, $(H_2)$, $(H_3)$ and $(H_4)$ hold for the weights $m, n$ and also for the weights $\hat{m}, \hat{n}$. If $m_1\leq \hat{m}_1$, $n_1\leq \hat{n}_1$, $\hat{m}_2\leq m_2$ and $\hat{n}_2\leq n_2$ a.e. in $\mathbb{R}^N$, then $c(m,n)\geq c(\hat{m},\hat{n})$. If in addition $$ \int_{\mathbb{R}^N}(\hat{m}-m)(u^+)^p + \int_{\mathbb{R}^N}(\hat{n}-n)(u^-)^p >0 $$ for at least one eigenfunction $u$ associated to $c(m,n)$, then $c(m,n)>c(\hat{m},\hat{n})$. \end{proposition} \begin{proof} Denote by $W$ (resp. $\hat{W}$) the weighted Sobolev space defined in Section 2 and associated to the weight $m_2$ (resp. $\hat{m}_2$). One clearly has $W\subset\hat{W}$. Once this has been observed, the proof is easily adapted from that of \cite[Propositions 23 and 25]{ari1}. One uses in particular the fact that if the path $\gamma$ is admissible in formula \eqref{b14} for $c(m,n)$, then $\gamma(t)\in \hat{W}$ for all $t\in[-1,1]$, and so by normalization one can construct a path in $\hat{W}$ which will be admissible in formula (\ref{b24'}) for $c(\hat{m},\hat{n})$. \end{proof} Let us also observe that definition \eqref{b14} clearly implies that $c(m,n)$ is homogeneous of degree $-1$: $c(sm,sn)=c(m,n)/s \text{ for } s>0$. Some sort of separate sub-homogeneity also holds, which will be useful later:\vspace{.3cm} \begin{proposition} \label{prop4.6} Assume that the weights $m, n$ satisfy hypothesis $(H_1)$, $(H_2)$, $(H_3)$ and $(H_4)$. If $00$ and take $\gamma\in\Gamma$ such that $\max_{t}J(\gamma(t)) < c(m,n)+\varepsilon$. Let $\gamma_k(t):= \gamma(t)/B_{m_k,n_k}(\gamma(t))^{1/p}$. We will show that $\gamma_k$ is well defined and that \begin{equation}\label{a1} \max_t J(\gamma_k(t)) < c(m,n) + \varepsilon \end{equation} for $k$ sufficiently large. Once this is done, one deduces from Proposition \ref{prop4.4} that $c(m_k,n_k)< c(m,n)+\varepsilon$ and consequently, since $\varepsilon>0$ is arbitrary, that $\limsup c(m_k,n_k)\leq c(m,n)$. The path $\gamma_k$ is clearly well defined if $B_{m_k,n_k}(\gamma(t))>0 \quad \forall t\in [0,1]$. To prove that this latter condition holds for $k$ sufficiently large, assume by contradiction that for a subsequence, $B_{m_k,n_k}(\gamma(t_k))\leq 0$ for some $t_k\in [0,1]$. For a further subsequence, $t_k\to t_0$ and $\gamma(t_k)\to \gamma(t_0)$ in $W$ and a.e. in $\mathbb{R}^N$. We claim that \begin{equation}\label{a2} B_{m,n}(\gamma(t_0))\leq 0, \end{equation} which is impossible since $\gamma\in\Gamma$. Deriving (\ref{a2}) is a matter of going to the limit in the expression $B_{m_k,n_k}(\gamma(t_k))\leq 0$. The ``$m_{1k}$ (or $n_{1k}$) terms" can be handled by H\"{o}lder inequality in a standard way. To handle the ``$m_{2k}$ (or $n_{2k}$) terms", we observe that $\gamma(t_k)\to\gamma(t_0)$ in $L^p(m_2, \mathbb{R}^N)$, which means that $m_2^{1/p}\gamma(t_k)\to m_2^{1/p}\gamma(t_0)$ in $L^p(\mathbb{R}^N)$, and consequently, for a further subsequence and for some $v\in L^p(\mathbb{R}^N)$, $|m_2^{1/p}\gamma(t_k)|\leq v$ a.e. in $\mathbb{R}^N$. This inequality and the fact that $m_{2k}$ and $n_{2k}$ are controlled by $m_2$ (a consequence of $(H_4)$ and (\ref{b24"})) allow the use of the dominated convergence theorem to handle these ``$m_{2k}$ (or $n_{2k}$) terms''. Let us now prove that (\ref{a1}) holds. Write $\max_t J(\gamma_k(t))= J(\gamma_k(\tau_k))$ and assume by contradiction that for a subsequence \begin{equation}\label{a3} J(\gamma_k(\tau_k))\geq c(m,n)+ \varepsilon. \end{equation} The preceding argument shows that for a further subsequence, one has $\tau_k\to \tau_0$ and $B_{m_k,n_k}(\gamma(\tau_k))\to B_{m,n}(\gamma(\tau_0))=1$. Consequently, by (\ref{a3}), $J(\gamma(\tau_0))\geq c(m,n)+ \varepsilon$, a contradiction with the choice of $\gamma$. To prove the lower semicontinuity, suppose by contradiction that for a subsequence, one has $c(m_k,n_k)\to c_0$ with $c_0< c(m,n)$. Let $u_k\in M_{m_k,n_k}$ be an eigenfunction associated to $c(m_k,n_k)$. We first show that $u_k$ remains bounded in $W$. One clearly has by the equation of $u_k$ that $\int_{\mathbb{R}^N}|\nabla u_k|^p = c(m_k,n_k)$ and so $\int_{\mathbb{R}^N}|\nabla u_k|^p $ remains bounded. To get a bound on $\int_{\mathbb{R}^N}m_2|u_k|^p $, one starts from $B_{m_k,n_k}(u_k)=1$. By using the imbeddings recalled in Section 2 (and Lemma \ref{lem2.2} when $N\leq p$), one easily sees that the ``$m_{1k}$ (and $n_{1k}$) terms" remain bounded. Replacing in the remaining terms $m_{2k}$ (resp. $n_{2k}$) by $m_2$ (resp. $n_2$) and using assumption $(H_4)$ and (\ref{b24"}), one deduces the desired bound on $\int_{\mathbb{R}^N}m_2|u_k|^p $. It follows that for a subsequence, $u_k\to u$ weakly in $W$, strongly in $L^p(m_1, \mathbb{R}^N)$ and in $L^p(n_1, \mathbb{R}^N)$. We will now prove that $u_k\to u$ in $W$. Taking $u_k-u_l$ as testing function in the equations satisfied by $u_k$ and by $u_l$, and writing $c(m_k,n_k)=c_k$, one has $$\int_{\mathbb{R}^N}|\nabla u_k|^{p-2}\nabla u_k\nabla (u_k-u_l) = c_k \int_{\mathbb{R}^N} [m_k(u_k^+)^{p-1}- n_k(u_k^-)^{p-1}](u_k-u_l),$$ which implies \begin{align*} 0&\leq \int_{\mathbb{R}^N}(|\nabla u_k|^{p-2}\nabla u_k- |\nabla u_l|^{p-2}\nabla u_l)\nabla (u_k-u_l)\\ &= c_k \int_{\mathbb{R}^N} m_k[(u_k^+)^{p-1}- (u_l^+)^{p-1}](u_k-u_l) \\ &\quad +c_k \int_{\mathbb{R}^N} n_k[-(u_k^-)^{p-1}+(u_l^-)^{p-1}](u_k-u_l)\\ &\quad+(c_k-c_l)\int_{\mathbb{R}^N}[m_k(u_l^+)^{p-1}-n_k(u_l^-)^{p-1}](u_k-u_l)\\ &\quad+c_l\int_{\mathbb{R}^N}[(m_k-m_l)(u_l^+)^{p-1}-(n_k-n_l)(u_l^-)^{p-1}](u_k-u_l)\\ &\leq c_k(I_1 + I_2) + |c_k-c_l|I_3 + c_l I_4 , \end{align*} where \begin{gather*} I_1= \int_{\mathbb{R}^N} m_{1k}[(u_k^+)^{p-1}- (u_l^+)^{p-1}](u_k-u_l), \\ I_2=\int_{\mathbb{R}^N} n_{1k}[-(u_k^-)^{p-1}+ (u_l^-)^{p-1}](u_k-u_l), \\ I_3= \int_{\mathbb{R}^N}[|m_k|(u_k^+)^{p-1}+|n_k|(u_l^-)^{p-1}]|u_k-u_l|, \\ I_4= \int_{\mathbb{R}^N}[|m_k-m_l|(u_l^+)^{p-1}+ |n_k-n_l|(u_l^-)^{p-1}]|u_k-u_l|. \end{gather*} Arguing as in the proof of Proposition \ref{prop3.3} , one then easily proves that $u_k\to u$ in $W$. (The full strength of the convergence (\ref{b24"}) is used here to verify that $I_4\to 0$). The limit $u\in M_{m,n}$ and is a solution of (\ref{b1}) for $\lambda=c_0$. Since $c_0< c(m,n)$, Theorem \ref{thm4.1} implies that $c_0=\lambda_1(m)$ and $u=\varphi_m $, or $c_0=\lambda_1(n)$ and $u=-\varphi_n$. Consider the first case (similar argument in the second one). Writing $v_k= u_k^-/(\int_{\mathbb{R}^N}|\nabla u_k^-|^p)^{1/p}$ and $\Omega_{k}^- = \{x\in \mathbb{R}^N: u_k(x)< 0\}$, we deduce from the equation satisfied by $u_k$ that $$ \frac{1}{c_k} = \int_{\mathbb{R}^N} n_k(v_k)^p\leq \int_{\Omega_k^-}n_{1k}(v_k)^p. $$ Consider the case $N>p$ (a similar argument holds in the case $N\leq p$). We will argue as in the proof of Proposition \ref{prop3.1}. By H\"{o}lder inequality one has $$ \frac{1}{c_k}\leq \|n_{1k}\|_{L^s(\Omega_k^-)} \|v_k\|_{L^{p^*}(\mathbb{R}^N)}^p\leq A^p \|n_{1k}\|_{L^s(\Omega_k^-)}, $$ which implies that for some $\varepsilon>0$, $\|n_{1k}\|_{L^s(\Omega_k^-)}\geq \varepsilon$ for $k$ sufficiently large. Moreover, since $n_{1k}\to n_1$ in $L^s(\mathbb{R}^N)$, one can choose $r>0$ so that $ \|n_{1k}\|_{L^s(B_r^c)}^s\leq \varepsilon^s/2$ for $k$ sufficiently large, and consequently $\|n_{1k}\|_{L^s(\Omega_{kr}^-)}^s\geq \varepsilon^s/2$ where $\Omega_{kr}^- := \Omega_k^-\cap B_r$. Since $n_{1k}$ converges to $n_1$ in $L_{\rm loc}^\infty(\mathbb{R}^N)$ one deduces from the latter relation that for some $\zeta>0$, $|\Omega_{kr}^-|>\zeta$ for $k$ sufficiently large. But this is impossible since $u_k\to \varphi_m$ in $L^p(B_r)$ and $\varphi_m(x)>0$ a.e. in $B_r$. \end{proof} Arguing as in Remark \ref{rmk2.3}, one can deduce from Proposition \ref{prop4.7} the following result. \begin{corollary} \label{coro4.8} Let $m=m_1-m_2, n=n_1-n_2$ satisfy $(H_1)$, $(H_2)_R$, $(H_3)$, $(H_4)_R$, and let $m_k=m_{1k}-m_{2k}, n_k=n_{1k}-n_{2k}$ with $m_{1k} , m_{2k} , n_{1k} , n_{2k}\geq 0$, $k=1,2,\dots $. Assume that $m_{1k}$, $n_{1k}$ belong to $L_{\rm loc}^\infty(\mathbb{R}^N)\cap L^s(\mathbb{R}^N)$ and converge in $L^s(\mathbb{R}^N)$ to $m_1$, $n_1$ respectively. Assume that $m_{2k}$, $n_{2k}$ belong to $L_{\rm loc}^\infty(\mathbb{R}^N)$ and converge in $L_{\rm loc}^\infty(\mathbb{R}^N)$ to $m_2$, $n_2$ respectively, with in addition, for some $\varepsilon_k\to 0$, \begin{equation} |m_{2k}-m_2|\leq \varepsilon_k m_2 \text{ and } |n_{2k}-n_2|\leq \varepsilon_kn_2\quad \text{ for a.e. $x$ with } |x|\geq R. \end{equation} Then $c(m_k, n_k)\to c(m,n)$. \end{corollary} \begin{remark} \label{rmk4.9} \rm If $(H_4)$ is replaced in Proposition \ref{prop4.7} by $(H_4')$ (cf. Remark \ref{rmk2.5}), then (\ref{b24"}) can be replaced by the natural requirement that $m_{2k},n_{2k}$ converge in $L^{N/p}(\mathbb{R}^N)$ to $m_2,n_2$ respectively. \end{remark} \section{Fu\v{c}ik spectrum in $\mathbb{R}^N$} The weights $m$ and $n$ in this section satisfy as before assumptions $(H_1)$, $(H_2)$, $(H_3)$ and $(H_4)$ (or $(H_4')$). The Fu\v{c}ik spectrum $\Sigma$ was defined in the introduction (cf. (\ref{b3})) and we will mainly consider here its part which lies in $\mathbb{R}^+\times \mathbb{R}^+$. This part clearly contains the half lines $\lambda_1(m)\times \mathbb{R}^+$ and $\mathbb{R}^+\times \lambda_1(n)$. These half lines are in fact exactly made of those $(\alpha, \beta)\in \Sigma\cap (\mathbb{R}^+\times \mathbb{R}^+)$ for which (\ref{b3}) admits a solution which does not change sign. We denote below by $\Sigma^*$ the set $\Sigma\cap (\mathbb{R}^+\times \mathbb{R}^+)$ without these 2 trivial half lines. From the properties of the first eigenvalue recalled in Section 2, it easily follows that if $(\alpha, \beta)\in \Sigma^*$, then $\alpha>\lambda_1(m)$ and $\beta>\lambda_1(n)$. \begin{theorem} \label{thm5.1} For any $r>0$, the line $\beta= r\alpha$ in the $(\alpha, \beta)$ plane intersects $\Sigma^*$. Moreover the first point in this intersection is given by $\alpha(r)= c(m,rn)$, $\beta(r)= r\alpha(r)$, where $c(.,.)$ is defined in \eqref{b14}. \end{theorem} The proof of the above theorem is an easy consequence of Theorem \ref{thm3.2} and Theorem \ref{thm4.1} Letting $r>0$ vary, we thus get a first curve $\mathcal{C}:=\{( \alpha(r), \beta(r) ):r>0\}$ in $\Sigma^*$. Here are some properties of this curve. \begin{proposition} \label{prop5.2} The functions $\alpha(r)$ and $\beta(r)$ in Theorem \ref{thm5.1} are continuous. Moreover $\alpha(r)$ is strictly decreasing and $\beta(r)$ is strictly increasing. One also has that $\alpha(r)\to +\infty$ if $r\to 0$ and $\beta(r)\to +\infty$ if $r\to +\infty$. \end{proposition} \begin{proof} The continuity of the functions $\alpha(r)$ and $\beta(r)$ follows directly from Proposition \ref{prop4.7}, and their strict monotonicity from Proposition \ref{prop4.6}. To show that $\alpha(r)\to +\infty$ as $r\to 0$, let us assume by contradiction that $\alpha(r)$ remains bounded as $r \to 0$. Then $\beta(r) = r\alpha(r)\to 0$ as $r\to 0$, which is impossible since $\beta(r)>\lambda_1(n)$ for all $r>0$. Similar argument for the behaviour of $\beta(r)$ as $r\to +\infty$. \end{proof} We now investigate the asymptotic values $\alpha_\infty:=\lim_{r \to +\infty}\alpha(r)$ and $\beta_\infty:=\lim_{r \to 0}\beta(r)$ of the first curve in $\mathbb{R}^+\times \mathbb{R}^+$. We will limit ourselves below to the study of $\alpha_\infty$; similar results can be proved for $\beta_\infty$ by interchanging $m$ and $n$. The following lemma will be used. It is concerned with the eigenvalue problem \begin{equation}\label{b25} -\Delta_pu = \lambda m|u|^{p-2}u \text{ in }B_R,\quad u=0 \text{ on } \partial B_R, \end{equation} where $B_R$ denotes the ball centred at the origin with radius $R$. Let $\lambda_R$ denote the first positive eigenvalue of (\ref{b25}) and $\varphi_R$ the associated positive eigenfunction such that $ \int_{B_R}m (\varphi_R)^p =1$ (which by $(H_3)$ clearly exist for $R$ sufficiently large ).\vspace{.2cm} \begin{lemma} \label{lem5.3} $\lambda_R \to \lambda_1(m)$ and $\varphi_R\to \varphi_m$ in $W$ as $R\to +\infty$. \end{lemma} \begin{proof} We will adapt some arguments from \cite[Lemma 5.2]{fig1}. The function $\varphi_R$ (extended by 0 outside $B_R$) clearly belongs to $W$ and satisfies $\int_{\mathbb{R}^N}m(\varphi_R)^p = 1$. This implies $$ \lambda_1(m)\leq \int_{\mathbb{R}^N} |\nabla \varphi_R|^p = \lambda_R , $$ and so $\liminf\lambda_R\geq \lambda_1(m)$. Let now $\delta\in ]0,1[$. Since $\varphi_m\in W$, there exists $\psi\in C_c^\infty(\mathbb{R}^N)$ such that \begin{gather*} \big|\int_{\mathbb{R}^N}(|\nabla \varphi_m|^p -|\nabla \psi|^p)\big|\leq \frac{\delta}{2}, \quad \big|\int_{\mathbb{R}^N} m_2(\varphi_m^p - |\psi|^p)\big| \leq \frac{\delta}{2}, \\ \big|\int_{\mathbb{R}^N}m_1(\varphi_m^p - |\psi|^p)\big| \leq \frac{\delta}{2}, \end{gather*} where we have used the imbedding of $W$ into $L^p(m_1, \mathbb{R}^N)$. This implies that $|\int_{\mathbb{R}^N}m(\varphi_m^p - |\psi|^p)|\leq \delta$ and since $\int_{\mathbb{R}^N} m|\varphi_m|^p =1$, one deduces that $\int_{\mathbb{R}^N} m|\psi|^p> 0$. We thus have, for $R$ sufficiently large, $$ \lambda_R\leq \frac{\int_{\mathbb{R}^N}|\nabla \psi|^p}{\int_{\mathbb{R}^N} m|\psi|^p} \leq \frac{\delta/2+\int_{\mathbb{R}^N}|\nabla \varphi_m|^p }{-\delta +\int_{\mathbb{R}^N}m|\varphi_m|^p}=\frac{\delta/2+\lambda_1(m)}{-\delta+1}, $$ and since $\delta>0$ is arbitrary, we conclude that $\lim\sup \lambda_R\leq \lambda_1(m)$. Let us now prove that $\varphi_R\to \varphi_m$ in $W$ as $R\to +\infty$. One has \begin{equation}\label{b27'} \int_{\mathbb{R}^N}|\nabla \varphi_R|^p =\lambda_R \to \lambda_1(m)= \int_{\mathbb{R}^N}|\nabla \varphi_m|^p, \end{equation} and so $ \int_{\mathbb{R}^N}|\nabla\varphi_R|^p$ remains bounded. Moreover using the imbeddings of Section 2 as well as Lemma \ref{lem2.2}, one deduces from $\int_{\mathbb{R}^N}m(\varphi_R)^p =1$ that $ \int_{\mathbb{R}^N}m_2(\varphi_R)^p$ remains bounded. Consequently $\varphi_R$ remains bounded in $W$, and for a subsequence, $\varphi_R\to v$ weakly in $W$ and strongly in $L^p(m_1, \mathbb{R}^N)$. One has $$ \int_{\mathbb{R}^N}|\nabla v|^p\leq \liminf \int_{\mathbb{R}^N}| \nabla \varphi_R|^p = \lambda_1(m), $$ and also $\int_{\mathbb{R}^N}m|v|^p\geq 1$ (where the latter inequality follows from $$ \int_{\mathbb{R}^N}m_2|v|^p\leq \liminf\int_{\mathbb{R}^N} m_2(\varphi_R)^p= -1+ \int_{\mathbb{R}^N}m_1|v|^p ). $$ The simplicity of $\lambda_1(m)$ then implies $v=\varphi_m$. One also has \begin{equation} \label{b27"} \lim\int_{\mathbb{R}^N} m_2 (\varphi_R)^p = \lim \Big[-1+ \int_{\mathbb{R}^N} m_1 (\varphi_R)^p\Big] = -1+\int_{\mathbb{R}^N}m_1\varphi_m^p= \int_{\mathbb{R}^N} m_2\varphi_m^p. \end{equation} Combining (\ref{b27'}) and (\ref{b27"}) with the weak convergence yields that $\varphi_R\to \varphi_m$ in $W$. \end{proof} The following proposition describes the asymptotic behaviour of the first curve $\mathcal{C}$. Let us recall that the support of a measurable function $u$ in $\mathbb{R}^N$ is defined as the complement in $\mathbb{R}^N$ of the largest open set on which $u=0$ a.e. \begin{proposition} \label{prop5.4} If $N\geq p$, then $\alpha_\infty = \lambda_1(m)$. If $N \lambda_1(m)$ if $\mathop{\rm supp} n^+$ is bounded. \end{proposition} \begin{proof} One starts by introducing \begin{equation}\label{b27} \overline{\alpha}:=\inf\left\{\int_{\mathbb{R}^N}|\nabla u^+|^p:u\in W,\int_{\mathbb{R}^N}m(u^+)^p=1 \text{ and } \int_{\mathbb{R}^N} n(u^-)^p >0\right\} \end{equation} and showing that $\alpha_\infty=\overline{\alpha}$. The proof of this equality is a direct adaptation of \cite{ari1}. One also clearly has $\overline{\alpha}\geq \lambda_1(m)$. We first consider the case $N\geq p$. In this case, the arguments of \cite{ari1}, which are local (they essentially involve approximating $\varphi_m$ by a function which vanishes on a small ball where $n^+\not\equiv 0$), can be easily adapted to the present situation and give $\overline{\alpha}=\lambda_1(m)$. We now consider the case where $N0$, $n^+\not\equiv 0$ on $\mathbb{R}^N\backslash\overline{B_R}$. This allows by a regularization procedure to construct $w_R\in C_c^\infty (\mathbb{R}^N\backslash\overline{B_R})$ with $w_R\geq 0$ and $\int_{\mathbb{R}^N}nw_R^p>0$. It follows that the function $$ u_R = \varphi_R - \frac{w_R}{R\|w_R\|_W} $$ converges to $\varphi_m$ in $W$ as $R\to +\infty$ and is admissible in the definition (\ref{b27}) of $\overline{\alpha}$. Since by Lemma \ref{lem5.3}, $\int_{\mathbb{R}^N}|\nabla \varphi_R|^p\to \lambda_1(m)$, we conclude that $\overline{\alpha}\leq \lambda_1(m)$, and so $\overline{\alpha} = \lambda_1(m)$. We finally consider the case where $N0$ on the compact set $\mathop{\rm supp} n^+$, it follows from the fact that $u_k^+\to\varphi_m$ uniformly on $\mathop{\rm supp} n^+$, that $ u_k^+\geq \varepsilon/2$ on $\mathop{\rm supp} n^+$ for $k$ sufficiently large. Consequently, for those $k$, $u_k^- = 0$ on $\mathop{\rm supp} n^+$, which implies $$ \int_{\mathbb{R}^N}n(u_k^-)^p = \int_{\mathbb{R}^N}n^+(u_k^-)^p -\int_{\mathbb{R}^N}n^-(u_k^-)^p= -\int_{\mathbb{R}^N}n^-(u_k^-)^p\leq 0 . $$ But this contradicts the fact that $u_k$ is admissible in the definition (\ref{b27}) of $\overline{\alpha}$. \end{proof} \begin{remark} \label{rmk5.5} \rm The distribution of $\Sigma$ in the other quadrants of $\mathbb{R}\times \mathbb{R}$ can be studied in a manner similar to that used in \cite{ari1}. It follows in particular that if $N>p$, $m,n\in L^{N/p}(\mathbb{R}^N)\cap L_{\rm loc}^\infty(\mathbb{R}^N)$, and $m$ and $n$ change sign, then $\Sigma$ contains a first hyperbolic-like curve in each quadrant. 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