\documentclass[reqno]{amsart} \usepackage{amssymb} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2006 International Conference in Honor of Jacqueline Fleckinger. \newline {\em Electronic Journal of Differential Equations}, Conference 16, 2007, pp. 81--93. \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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{81} \title[\hfilneg EJDE/Conf/16 \hfil Infinite-dimensional nonlinear elliptic problems] {Some remarks on infinite-dimensional nonlinear elliptic problems} \author[Ph. Cl\'ement, M. Garc\'{\i}a-Huidobro, and R. Man\'asevich \hfil EJDE/Conf/16 \hfilneg] {Philippe Cl\'ement, Marta Garc\'{\i}a-Huidobro, Ra\'ul F. Man\'asevich} % in alphabetical order \address{Philippe Cl\'ement \newline Mathematical Institute, Leiden University, P.O. Box 9512, NL-2300, RA Leiden, and EEMCS/ DIAM, TU Delft, P.O. Box 5031, NL-2600, GA Delft, The Netherlands} \email{clement@math.leidenuniv.nl} \address{Marta Garc\'{\i}a-Huidobro \newline Departamento de Matem\'aticas, Pontificia Universidad Cat\'olica de Chile, Casilla 306, Correo 22, Santiago, Chile} \email{mgarcia@mat.puc.cl} \address{Ra\'ul F. Man\'asevich \newline Departamento de Ingenier\'{\i}a Matem\'atica and Centro de Modelamiento Matem\'atico, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile} \email{manasevi@dim.uchile.cl} \dedicatory{Dedicated to Jacqueline Fleckinger on the occasion of\\ an international conference in her honor} \thanks{Published May 15, 2007.} \subjclass[2000]{35J65, 35J25} \keywords{Hilbert space; Ornstein-Uhlenbeck operator; \hfill\break\indent nonlinear elliptic problems} \thanks{Ph. Cl\'ement was supported by grant 7150117 from FONDECYT; M. Garc\'{\i}a-H. by \hfill\break\indent grant 1030593 from FONDECYT; R. Man\'asevich by grant P04-066-F from Fondap \hfill\break\indent Matem\'aticas Aplicadas and Milenio.} \begin{abstract} We discuss some nonlinear problems associated with an infinite dimensional operator $L$ defined on a real separable Hilbert space $H$. As the operator $L$ we choose the Ornstein-Uhlenbeck operator induced by a centered Gaussian measure $\mu$ with covariance operator $Q$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \section{Introduction} The goal of this note is to present some results for nonlinear problems associated with an infinite dimensional operator $L$ defined on a real separable Hilbert space $H$. As the operator $L$ we choose the Ornstein-Uhlenbeck operator induced by a centered Gaussian measure $\mu$ with covariance operator $Q$ (see \cite{daP}). In the first part we consider existence and uniqueness of solutions for a problem of the form \begin{equation}\label{11} -Lu+\beta(u)=f, \end{equation} where $\beta$ satisfies \begin{itemize} \item[(H1)] %H\beta $\beta$ is a strictly increasing homeomorphism of $\mathbb{R}$ onto $\mathbb{R}$, $\beta(0)=0$, \end{itemize} and $f\in L^2(H,\mu)$ is given. As a consequence of the existence part we can show that the operator $L(\beta^{-1})$, with an appropriate domain, has an $m$-dissipative closure in $L^1(H,\mu)$. Thus, in view of the Crandall-Liggett Theorem, see \cite{CrL} (and also \cite{Cr2}), it generates a nonlinear contraction semigroup on the closure of its domain in $ L^1(H,\mu)$. In the second part we make the additional assumption that $\beta$ is odd and we consider the nonlinear eigenvalue problem \begin{equation}\label{12} -Lu+\beta(u)=\lambda u,\quad \lambda\ge 0, \end{equation} where $u\in L^2(H,\mu)$, $\|u\|_{ L^2(H,\mu)}=R$, with $R>0$ given. By using results in \cite{Cl1} and \cite{K}, we obtain the existence of an infinite sequence $\{(\lambda_n,u_n)\}_{n\in\mathbb{N}}$ of solutions to \eqref{12} with $\lambda_n\to\infty$ as $n\to\infty$. This implies the existence of infinitely many solution pairs $(\lambda,u)$ with non constant $u$. Moreover, we discuss the existence of solutions with nonnegative and non constant $u$. \section{Preliminaries} In this section we establish the notation that we will use throughout this work. Most of it is taken from \cite{daP} and we refer the reader to this book. $H$ will denote a finite or infinite dimensional real separable Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $|\cdot|$. Throughout the paper $\mu=N_Q$ will denote the centered Gaussian measure on $H$ with covariance $Q$, (see \cite[page 12]{daP}), where $Q$ denotes a positive symmetric operator of trace class in $H$ with $Ker(Q)=\{0\}$. Also, $\{e_k\}_{k\in\mathbb{N}}$ will denote a complete orthonormal system of eigenvectors of $Q$ with corresponding eigenvalues $\{\gamma_k\}_{k\in\mathbb{N}}$ satisfying \begin{equation}\label{eigen} 0<\gamma_{k+1}\le\gamma_k. \end{equation} We recall here that the Ornstein-Uhlenbeck semigroup \lq\lq associated with $\mu$\rq\rq\ is given by $$ R_t\varphi(x)=\int_H\varphi(e^{tA}x+y)N_{Q_t}(dy), \quad x\in H,\; t>0, $$ and $\varphi\in B_b(H)$ (Borel bounded functions on $H$). Here $A=-\frac{1}{2}Q^{-1}$ and $$ Q_tx=\int_0^te^{2sA}xds=Q(I-e^{2tA})x,\quad x\in H,\; t>0. $$ As a consequence of \cite[Proposition 10.22]{daP}, $R_t$ can be uniquely extended to a strongly continuous contraction semigroup in $ L^2(H,\mu)$, which we still denote by $R_t$, and $\mu$ is the unique invariant measure of $R_t$ and for $x\in H$, $$ \lim_{t\to\infty}R_t\varphi(x)=\int_H\varphi(y)d\mu(y)=\overline{\varphi}. $$ Moreover, from \cite[Th5.8]{daP}, $R_t$ can be uniquely extended to a strongly continuous positive contraction semigroup in $ L^p(H,\mu)$ for all $1\le p<\infty$. We shall denote by $L_p$ the infinitesimal generator of $R_t$ in $ L^p(H,\mu)$. In particular, $L_1$ is $m$-dissipative in $ L^1(H,\mu)$ hence it satisfies \begin{equation}\label{21} \int_H(L_1u)(x)\mathop{\rm sgn}(u(x))d\mu\le 0,\quad \mbox{for every }u\in D(L_1), \end{equation} see e.g. \cite[Lemma 2]{BS}, where we have used the notation $$ \mathop{\rm sgn}(t)=\begin{cases}1& t>0,\\ 0 & t=0,\\ -1 &t<0.\end{cases} $$ Moreover, $-L_2$ is a nonnegative self adjoint operator in $ L^2(H,\mu)$ with domain \begin{equation}\label{domL2} D(L_2)=\{u\in W^{2,2}(H,\mu):\int_H|(-A)^{1/2}D u|^2d\mu<\infty\}, \end{equation} see \cite[Propositions 10.22 and 10.34]{daP} and \begin{equation}\label{defL2} L_2\varphi(x)=\frac{1}{2}\mbox{Tr}[D^2\varphi](x)+\langle x,AD\varphi(x)\rangle, \end{equation} where $\varphi\in\mathcal{E}_A(H)$, which is defined to be the linear span in $C_b(H)$ (continuous bounded functions in $H$) of real and imaginary parts of $\varphi_h$, where $\varphi_h(x)=e^{i\langle h,x\rangle}$, $h\in D(A)$, and $D$, $D^2$ are the differential operators introduced in \cite[Proposition 10.3 and 10.32]{daP}. We also introduce $\mathcal{E}(H)$ as the linear span in $C_b(H)$ of real and imaginary parts of $\varphi_h$, where $\varphi_h(x)=e^{i\langle h,x\rangle}$, and now $h\in H$. Finally we note also that the null space $N(L_p)=\{{\rm const.}\}$, $1\le p<\infty$. We also consider the Dirichlet form $a: W^{1,2}(H,\mu)\times W^{1,2}(H,\mu)\to\mathbb{R}$ defined by $$ a(\varphi,\psi)=\frac{1}{2}\int_H\langle D\varphi,D\psi\rangle d\mu.$$ The linear space $ W^{1,2}(H,\mu)$ endowed with the inner product $$ \langle \varphi,\psi\rangle_{ W^{1,2}(H,\mu)}=\langle\varphi,\psi\rangle_{ L^2(H,\mu)}+2a(\varphi,\psi) $$ is a real separable Hilbert space with \begin{equation}\label{compactimbe} W^{1,2}(H,\mu)\hookrightarrow L^2(H,\mu)\quad\mbox{compact}, \end{equation} see \cite[Theorem 10.16]{daP}. Finally, we recall that \begin{equation}\label{intbyparts} a(\varphi,\psi)=-\int_H\langle L_2\varphi,\psi\rangle d\mu \end{equation} for all $\varphi\in D(L_2),$ and all $\psi\in W^{1,2}(H,\mu)$, see \cite[Section 10.4]{daP}. \begin{remark}\label{remph} \rm We want to note that in this work the operator $L_2$ is defined as the generator of the semigroup $R_t$ in $ L^2(H,\mu)$, while in \cite{daP} the operator $L_2$ is defined on page 151 via the Lax-Milgram Theorem. In view of \cite[Proposition 10.22 (iv)]{daP}, they are the same. \end{remark} \section{An infinite dimensional porous media type operator} The aim of this section is to construct an infinite dimensional nonlinear second order elliptic operator which is of porous media type $\Delta(\beta^{-1})$, following the approach of \cite{Cr1} Let $\beta$ satisfy (H1) and consider problem \eqref{11}. \begin{proposition}\label{prop31} (a) For every $f\in L^2(H,\mu)$ there exists a unique $u\in D(L_2)$ such that $\beta(u)\in L^2(H,\mu)$ and $u$ satisfies \eqref{11} with $L=L_2$. \noindent(b) If \begin{itemize} \item[(H2)] %H_\gamma $\beta(u)=\varepsilon u+\gamma(u)$ for some $\varepsilon>0$ and some continuous monotone increasing function $\gamma:\mathbb{R}\to\mathbb{R}$, $\gamma(0)=0$, \end{itemize} then for any $f\in L^1(H,\mu)$ there exists a unique $u\in D(L_1)$ with $\beta(u)\in L^1(H,\mu)$ satisfying \eqref{11} with $L=L_1$. \end{proposition} \begin{proof} We start by proving $(a)$. Set $A:=-L_2$, $Bu(x):=\beta(u(x))$, where $$D(B)=\{u\in L^2(H,\mu):\beta(u)\in L^2(H,\mu)\}$$ and write \eqref{11} as $$ Au+Bu=f,\quad f\in L^2(H,\mu). $$ We claim that $A$ is maximal monotone and that it is the subdifferential of the convex l.s.c functional $J_a: L^2(H,\mu)\mapsto[0, \infty]$ defined by \begin{equation}\label{defJa} J_a(\varphi)=\begin{cases} a(\varphi,\varphi),& \varphi\in W^{1,2}(H,\mu),\\ +\infty &\mbox{otherwise.} \end{cases} \end{equation} Indeed, for $u\in D(L_2)$ and $h\in W^{1,2}(H,\mu)$, by \eqref{intbyparts}, we have that \begin{align*} J_a(u+h)&=J_a(u)+\int_H\langle Du,Dh\rangle d\mu+J_a(h)\\ &\ge J_a(u)-\int_H\langle L_2u,h\rangle d\mu, \end{align*} which implies that $u\in D(\partial J_a)$ and $-L_2u\in \partial J_a(u)$. Note that since $J_a$ is convex, it follows that $\partial J_a$ is monotone, moreover, since $-L_2$ is nonnegative and selfadjoint in $ L^2(H,\mu)$, it follows that it is maximal monotone. Hence $-L_2=\partial J_a$ by the maximal monotonicity of $-L_2$. Also, $B$ is the subdifferential of $$ J_b(u)=\begin{cases}\int_Hb(u)d\mu,&\text{if } \int_Hb(u)d\mu<\infty\\ \infty&\text{otherwise},\end{cases} $$ where \begin{equation}\label{defb} b(t):=\int_0^t\beta(s)ds. \end{equation} Therefore $A$ and $B$ satisfy all the assumptions in \cite[Example 1]{BH} implying that $$ \mathop{\rm Int}(R(A+B))=\mathop{\rm Int}(R(A)+R(B)). $$ Since $R(B)= L^2(H,\mu)$, we conclude that $R(A+B)= L^2(H,\mu)$. Finally, the uniqueness assertion follows from the strict monotonicity of $\beta$. Next we prove $(b)$. In order to achieve this we write \eqref{11} as \begin{equation}\label{11new} (\varepsilon-L_1)u+\gamma(u)=f,\quad f\in L^1(H,\mu). \end{equation} Hence in view of Theorem 1 in \cite{BS} it is sufficient to see that the operator $A:=\varepsilon-L_1$ satisfies $(I)$, $(II)$, and $(III)$ in \cite{BS}. Since by definition $L_1$ generates a linear contraction $C_0$ semigroup in $ L^1(H,\mu)$, so does $L_1-\varepsilon=-A$, which yields $(I)$. Also, from the dissipativity of $L_1$ we have that $$ \varepsilon\|u\|_{ L^1(H,\mu)}\le \|\varepsilon u-L_1\|_{ L^1(H,\mu)}=\|A\|_{ L^1(H,\mu)}, $$ implying that $(III)$ is also satisfied. Finally we prove $(II)$. Let $\lambda>0$ and $f\in L^1(H,\mu)$. Since the semigroup generated by $L_1$ is positive, we have that $$ (I+\lambda A)^{-1}f\le (I+\lambda A)^{-1}f^+ $$ and hence \begin{equation}\label{fdes2} \mathop{\rm ess\,sup}(I+\lambda A)^{-1}f\le \mathop{\rm ess\,sup}(I+\lambda A)^{-1}f^+. \end{equation} Since $L_p$ generates a linear contraction $C_0$ semigroup in $ L^p(H,\mu)$ for all $1\le p<\infty$, so does $L_p-\varepsilon$, hence \begin{equation}\label{fdes}\|(I+\lambda A)^{-1}f^+\|_{ L^p(H,\mu)}\le \|f^+\|_{ L^p(H,\mu)}, \end{equation} provided that $f^+\in L^p(H,\mu)$. Assuming $f^+\in L^\infty(H,\mu)$, by letting $p\to\infty$ in \eqref{fdes} we obtain \begin{align*} \mathop{\rm ess\,sup}(I+\lambda A)^{-1}f^+ &=\|(I+\lambda A)^{-1}f^+\|_{L^\infty(H,\mu)}\\ &\le \|f^+\|_{L^\infty(H,\mu)}= \mathop{\rm ess\,sup}f^+\\ &=\max\{0,\mathop{\rm ess\,sup}f\}. \end{align*} Therefore, using \eqref{fdes2} we conclude that $$ \mathop{\rm ess\,sup}(I+\lambda A)^{-1}f\le\max\{0,\mathop{\rm ess\,sup}f\}, $$ which is exactly assumption $(II)$ in \cite{BS}. \end{proof} We are now in a position to define a \lq\lq porous media type\rq\rq\ operator, which we denote by $L_\phi$, where $\phi=\beta^{-1}$ in $ L^1(H,\mu)$: $$ D(L_\Phi):=\{u\in L^1(H,\mu):\phi(u)\in D(L_1)\}, $$ and for $u\in D(L_\Phi)$ we set $$ L_\phi u:=L_1(\phi(u)). $$ We have the following result. \begin{theorem} \label{th31} \begin{itemize} \item[(i)] The closure of $L_\phi$ is a nonlinear (possibly multivalued) $m$-dissipative operator in $ L^1(H,\mu)$. \item[(ii)]If $\beta$ satisfies assumption (H2), then $L_\phi$ is itself a nonlinear $m$-dissipative operator in $ L^1(H,\mu)$. \item[(iii)] If $\phi\in C^2(\mathbb{R})$, then $\overline{D(L_\phi)}= L^1(H,\mu)$. \end{itemize} \end{theorem} \begin{remark}\label{rem1} \rm We do not claim that the last two assertions in Theorem \ref{th31} are optimal. \end{remark} \begin{proof}[Proof of Theorem \ref{th31}] (i) We will first prove the dissipativity of $L_\phi$ in $ L^1(H,\mu)$. Let $u,\ v\in D(L_\phi)$ and let $\bar u=\phi(u)$, $\bar v=\phi(v)$. By assumption, $\bar u$ and $\bar v$ belong to $D(L_1)$. In view of the dissipativity of $L_1$ in $ L^1(H,\mu)$ we have \begin{equation}\label{diss} \int_H L_1(\bar u-\bar v)\mathop{\rm sgn}(\bar u-\bar v)d\mu\le 0, \end{equation} and in view of (H1), \begin{equation}\label{sign} \mathop{\rm sgn}( u- v)=\mathop{\rm sgn}(\bar u-\bar v). \end{equation} Hence, replacing \eqref{sign} into \eqref{diss}, and using the definition of $\bar u,\ \bar v$ we get $$ \int_H(L_1(\phi(u)-\phi(v))\mathop{\rm sgn}(u- v)d\mu\le 0, $$ which implies the dissipativity of $L_\phi$. We prove now that $R(I- L_\phi)$ is dense in $ L^1(H,\mu)$. Let $f\in L^2(H,\mu)$. Then by Proposition \ref{prop31} (a), there exists $v\in D(L_2)$, with $\beta(v)\in L^2(H,\mu)$, such that $$ -L_2v+\beta(v)=f, $$ hence setting $u=\beta(v)$ we obtain $v=\phi(u)$ and $$ u-L_2\phi(u)=f, $$ hence $f\in R(I-L_\phi)$ (since $L_2\subset L_1$). We conclude that $ L^2(H,\mu)\subseteq R(I-\lambda L_\phi)$ and the claim follows from the density of $ L^2(H,\mu)$ in $ L^1(H,\mu)$. It is a well known fact that if $\overline{L_\phi}$ denotes the closure of $L_\phi$, then $\overline{L_\phi}$ is dissipative (possibly multivalued) and $R(I-\overline{L_\phi})$ is closed, hence equal to $ L^1(H,\mu)$. Therefore $\overline{L_\phi}$ is $m$-dissipative in $ L^1(H,\mu)$. \noindent (ii) It follows from Proposition \ref{prop31} that if $\beta$ is of the form (H2) then the range $$ R(I-L_\phi)= L^1(H,\mu), $$ hence in this case $L_\phi$ is $m$-dissipative. \noindent (iii) It is sufficient to show that $\mathcal{E}_A(H)\subseteq D(L_\phi)$, since $\mathcal{E}_A(H)$ is dense in $ L^2(H,\mu)$. If $v\in\mathcal{E}_A(H)$, then there exists $N\ge 1$, $h_1,h_2,\ldots,h_N, k_1,k_2,\ldots,k_N\in D(A)$ $\alpha_1,\alpha_2,\ldots,\alpha_N,\beta_1,\beta_2,\ldots,\beta_N\in\mathbb{R}$ such that \begin{equation}\label{defv} v(x)=\sum_{i=1}^N\Bigl(\alpha_i\cos\langle h_i,x\rangle+\beta_i\sin\langle k_i,x\rangle\Bigr),\quad x\in H. \end{equation} We will prove that $\phi(v)\in D(L_2)$. In view of \eqref{domL2}, with first verify that $\phi(v)\in W^{2,2}(H,\mu)$. Since $v\in C_b(H)$, we have that $\phi(v),\phi'(v)$ and $\phi''(v)$ are in $C_b(H)$. In particular, $\phi(v)\in L^2(H,\mu)$. From the definition of $ W^{2,2}(H,\mu)$ in \cite[Section 10.6, page 161]{daP}, we need to compute $D_jD_\ell\phi(v)$, $j,\ell\in\mathbb{N}$. Since $D_jv$ and $D_\ell v$ are bounded and continuous, from $$ D_\ell\phi(v)=\phi'(v)D_\ell v, $$ and \begin{equation}\label{der2} D_jD_\ell\phi(v)(x)=\phi'(v)D_jD_\ell v(x)+ \phi''(v)D_jv(x)D_\ell v(x),\end{equation} we obtain that $D_jD_\ell\phi(v)\in C_b(H)\subseteq L^2(H,\mu)$. Next we show that \begin{equation}\label{sum2} \sum_{j,\ell=1}^\infty\int_H|D_jD_\ell\phi(v)|^2d\mu<\infty. \end{equation} From \eqref{der2} it is sufficient to show that \begin{equation}\label{2sumas} \sum_{j,\ell=1}^\infty\int_H|D_jD_\ell v|^2d\mu<\infty\quad\mbox{and}\quad \sum_{j,\ell=1}^\infty\int_H|D_jv(x)D_\ell v(x)|^2d\mu<\infty. \end{equation} Indeed, the first assertion in \eqref{2sumas} follows from the fact that $v\in\mathcal{E}_A(H)\subseteq\mathcal{E}(H)\subseteq W^{2,2}(H,\mu)$. For the second one we note that \begin{equation}\label{dj} |D_jv(x)|\le C\sum_{i=1}^N(|\langle h_i,e_j\rangle|+|\langle k_i,e_j\rangle|) \end{equation} where $C$ is a positive constant depending only on $\alpha_1,\ldots,\alpha_N,\beta_1,\ldots,\beta_N$, hence \begin{equation}\label{m1} \sum_{j,\ell=1}^\infty|D_jv(x)D_\ell v(x)|^2\le 4N^2C^4 \Bigl(\sum_{i=1}^N|h_i|^2+|k_i|^2\Bigr)^2, \end{equation} implying that the second assertion in \eqref{2sumas} holds and therefore $\phi(v)\in W^{2,2}(H,\mu)$. Finally we will prove that \begin{equation}\label{toprove} \int_H|(-A)^{1/2}D \phi(v)|^2d\mu<\infty. \end{equation} First we prove that $D\phi(v)(x)\in D((-A)^{1/2})$. Since $A=-\frac{1}{2}Q^{-1}$, $w\in H$ belongs to $D((-A)^{1/2})$ if and only if \begin{equation}\label{dom} \sum_{j=1}^\infty\gamma^{-1}_j\langle w,e_j\rangle^2<\infty. \end{equation} Now, $D\phi(v)(x)=\phi'(v)D_jv(x)$ and $|\phi'(v)|\le C_0$ for some positive constant $C_0$, hence from \eqref{dj} we find that $$ |D_j\phi(v)(x)|^2\le C_0^2|D_jv(x)|^2\le 2NC_0^2C^2 \sum_{i=1}^N(|\langle h_i,e_j\rangle|^2+|\langle k_i,e_j\rangle|^2) $$ where $h_i,\ k_i\in D(A)$, $i=1,\ldots,N$, that is, \begin{equation}\label{1point} \sum_{j=1}^\infty\gamma_j^{-2}|\langle h_i,e_j\rangle|^2<\infty\quad\mbox{and}\quad \sum_{j=1}^\infty\gamma_j^{-2}|\langle k_i,e_j\rangle|^2<\infty. \end{equation} Hence from \eqref{1point}, $$ \sum_{j=1}^\infty\gamma_j^{-1}|D_j\phi(v)(x)|^2\le2NC_0^2C^2 \sum_{i=1}^N\sum_{j=1}^\infty\gamma_j^{-1} (|\langle h_i,e_j\rangle|^2+|\langle k_i,e_j\rangle|^2)<\infty, $$ since by \eqref{eigen} $\gamma_j^{-1}\le \gamma_j^{-2}$ for large $j$. This implies that $D\phi(v)(x)\in D((-A)^{1/2})$ for any $x\in H$ and \begin{align*} |(-A)^{1/2}D\phi(v)|^2(x) &=\sum_{j=1}^\infty\langle D\phi(v)(x),(-A)^{1/2}e_j\rangle^2\\ &=\frac{1}{2}\sum_{j=1}^\infty\langle D\phi(v)(x),\gamma_j^{-1/2}e_j\rangle^2\\ &=\frac{1}{2}\sum_{j=1}^\infty\gamma_j^{-1}\langle D\phi(v)(x),e_j\rangle^2, \end{align*} implying that the integrand in \eqref{toprove} is Borel measurable and bounded and thus \eqref{toprove} holds. This completes the proof of part (3). \end{proof} We end this section by giving some properties of the nonlinear semigroup generated by $\overline{L_\phi}$. \begin{proposition}\label{prop32} Let $\beta$ satisfy (H1) and $S_t:\overline{D(\overline{L_\phi})}\to \overline{D(\overline{L_\phi})}$ be the nonlinear semigroup generated by $\overline{L_\phi}$. Then the following hold. \begin{itemize} \item[(i)] For any $c\in\mathbb{R}$, $c\in D(L_\phi)$, and $S_t(c)=c$. \item[(ii)] Let $f,\ g\in \overline{D(\overline{L_\phi})}$ such that $f\le g$. Then $S_t(f)\le S_t(g)$ for all $t>0$. \item[(iii)] For any $f\in \overline{D(\overline{L_\phi})}$, $$\int_HS_tfd\mu=\int_Hfd\mu\quad\mbox{for all }t>0.$$ \end{itemize} \end{proposition} \begin{proof} From Proposition \ref{prop31}, for any $h>0$ there is a unique $u\in L^2(H,\mu)$ such that \begin{equation}\label{mgh1} -L_2u+\frac{1}{h}\beta(u)=\frac{1}{h}f, \end{equation} hence \begin{equation}\label{mgh2} (I-h\overline{L_\phi}))^{-1}f=\beta(u). \end{equation} \noindent Proof of (i). If $f=c$, we have $\beta(u)=c$ and thus by induction it follows that \begin{equation}\label{i1} (I-h\overline{L_\phi})^{-m}c=c\quad\mbox{for all }m\in\mathbb{N}, \end{equation} therefore, for any $t>0$ we have $$ S_t(c)=\lim_{m\to\infty}(I-\frac{t}{m}\overline{L_\phi})^{-m}c=c $$ \noindent Proof of (ii). Let now $f_1,\ f_2\in L^2(H,\mu)$, with $f_1\le f_2$, and for $h>0$ and $\varepsilon >0$, and $i=1,2$, let $u_i^{\varepsilon}$ satisfy $$ \varepsilon u_i^{\varepsilon}-L_2 u_i^{\varepsilon} +\frac{1}{h}\beta(u_i^{\varepsilon})=\frac{1}{h}f_1. $$ From \cite[Proposition 4.7 (iv) implies (i)]{Br-book} with $$ \varphi(u)=\begin{cases} 0 & u\ge 0\\ +\infty &\mbox{otherwise},\end{cases} $$ we obtain $u_1^{\varepsilon}\le u_2^{\varepsilon}$. By letting $\varepsilon\to0$ we obtain $u_1\le u_2$ where $u_i$ satisfy $$ -L_2u_i+\frac{1}{h}\beta(u_i)=f_i,\quad i=1,2. $$ Hence, $\beta(u_1)\le\beta(u_2)$ and thus $$ (I-h\overline{L_\phi}))^{-1}f_1\le (I-h\overline{L_\phi}))^{-1}f_2. $$ Therefore, by induction, \begin{equation}\label{i2} (I-h\overline{L_\phi}))^{-m}f_1\le (I-h\overline{L_\phi}))^{-m}f_2. \end{equation} Since $ L^2(H,\mu)$ is dense in $ L^1(H,\mu)$, \eqref{i2} holds also for $f_1,\ f_2\in L^1(H,\mu)$. By taking $f_1,\ f_2\in \overline{D(\overline{L_\phi})}$, we obtain as before that $S_t(f_1)\le S_t(f_2)$. \noindent Proof of (iii). Arguing as before, it is sufficient to prove that $$\int_H(I-h\overline{L_\phi})^{-1}fd\mu=\int_Hfd\mu$$ for all $h>0$ and $f\in L^2(H,\mu)$. This follows by integrating \eqref{mgh1} over $H$ to obtain $$ \int_H\beta(u)d\mu=\int_Hfd\mu, $$ hence our claim follows by integrating now \eqref{mgh2} over $H$. \end{proof} \section{A nonlinear eigenvalue problem associated with the Ornstein-Uhlenbeck operator} In this section we consider the nonlinear eigenvalue problem \begin{equation}\label{41} -L_2u+\beta(u)=\lambda u, \end{equation} where $\beta$ satisfies (H1) and is odd. By a solution to this equation we mean a pair $(\lambda,u)\in\mathbb{R}\times L^2(H,\mu)$ satisfying $u\in W^{2,2}(H,\mu)$, $\beta(u)\in L^2(H,\mu)$. Clearly, for any $\lambda\in\mathbb{R}$, $(\lambda,0)$ is a solution to \eqref{41}. Set $$ \lambda^*:=\sup\{\lambda\in\mathbb{R}:s\mapsto\beta(s) -\lambda s\mbox{ is strictly increasing in }\mathbb{R}\} $$ We have that $0\le \lambda^*<\infty$. If $\lambda<\lambda^*$, then $s\mapsto \beta(s)-\lambda s$ is strictly increasing and hence, from Proposition \ref{prop31} (a) we have that $(\lambda,0)$ is the only solution to \eqref{41}. For $\lambda\in\mathbb{R}$ let us consider the functional $J_\lambda: L^2(H,\mu)\to[-\infty,\infty]$ defined by \begin{equation}\label{defJl} \qquad J_\lambda(u)=\begin{cases} J_a(u)+J_b(u)-\frac{\lambda}{2}\|u\|_{ L^2(H,\mu)}^2, & u\in W^{1,2}(H,\mu),\; \int_Hb(u)d\mu<\infty\\ +\infty &\mbox{otherwise.}\end{cases} \end{equation} We observe that for $\lambda<\lambda^*$, $J_\lambda$ is strictly convex, l.s.c. and nonnegative, and $0$ is its global minimizer. Next we investigate the positive constant solutions to \eqref{41} $u(x)\equiv c$. Then $\beta(c)=\lambda c$. We have the following result. \begin{proposition}\label{prop41} Assume that \begin{equation}\label{beta-extra1} t\mapsto \beta(t)/t\mbox{ is strictly increasing on }(0,\infty). \end{equation} Then for all $c>0$ the pair $(\beta(c)/c,c)$ is a solution to \eqref{41} and $u= c$ minimizes the functional $J_0$ on the set $$ S_c:=\{u\in W^{1,2}(H,\mu):\|u\|_{ L^2(H,\mu)}=c\}. $$ Furthermore, $u= c$ is the unique nonnegative minimizer of $J_0$ on $S_c$. \end{proposition} \begin{proof} From \eqref{beta-extra1} we obtain that the mapping $t\mapsto b(\sqrt{t})$, $t>0$, is strictly convex, hence for any $u\in D(J_0)$ we have by Jensen's inequality (\cite[Theorem 2.2(a)]{LL}) that \begin{equation}\label{ph1} J_0(u)\ge\int_Hb(\sqrt{|u|^2})d\mu\ge b\Bigl(\sqrt{\int_H|u|^2d\mu}\Bigr) =b(c)=J_0(c), \end{equation} implying that $u= c$ is a minimizer for $J_0$. On the other hand, if $u$ is a minimizer, then from \eqref{ph1} and the fact that $J_0(c)\ge J_0(u)$, we obtain that $$ \int_Hb(\sqrt{|u|^2})d\mu= b\Bigl(\sqrt{\int_H|u|^2d\mu}\Bigr), $$ hence by \cite[Theorem 2.2(b)]{LL} we deduce that $u^2$ must be a constant, hence $u= c$ since $u$ is nonnegative. \end{proof} We will now state and prove our existence results. \begin{theorem}\label{existprop} \begin{itemize} \item[(i)]For any $R>0$ there exists a solution $(\lambda,u)$ to \eqref{41} satisfying $u\ge 0$ and $u$ minimizes $J_0$ on $S_R$. \item[(ii)]For any $R>0$ there exists a sequence of solutions $\{(\lambda_n,u_n)\}_{n\in\mathbb{N}}$ to \eqref{41} such that $u_n\in S_R$ and \begin{equation}\label{eigenprop} \lambda_n>0\quad\mbox{for } n\in\mathbb{N},\quad\mbox{and}\quad \sup_{n\in\mathbb{N}}\lambda_n=\infty. \end{equation} \end{itemize} \end{theorem} \begin{proof} (ii) We will apply Theorem 1 in \cite{Cl1}, see also \cite{K}. As the real infinite dimensional separable Hilbert space we choose $E= L^2(H,\mu)$. Let $\varphi:E\to[0,\infty]$ be defined by $\varphi(u):=J_{-1},$ $u\in E$. Then clearly $\varphi$ is convex, even, and $\varphi(0)=0$. Moreover, in view of the compactness of the imbedding \eqref{compactimbe}, the convex set $$ \{u\in E:\varphi(u)\le \rho\} $$ is compact in $E$ for all $\rho\ge 0$. Moreover, since $\mathcal{E}(H)\subseteq C_b(H)\cap W^{1,2}(H,\mu)$ we have that $\mathcal{E}(H)\subseteq D(\varphi)$. The density of $\mathcal{E}(H)$ in $E$ implies the density of $D(\varphi)$ in $E$. Hence, all the assumptions of \cite[Theorem 1]{Cl1} are satisfied and therefore there exists a sequence $(\nu_k,u_k)\in\mathbb{R}\times E$, $k\in\mathbb{N}$ such that $\|u_k\|_{E}=R$, $\partial J_{-1}(u_k)\ni\nu_k u_k$ and $\sup_{k\ge 1}\varphi(u_k)=\infty$. We claim that $$ D(\partial J_{-1})=D(L_2)\cap D(B), $$ and $$ \partial J_{-1}(u)=-L_2u+Bu+u,\quad u\in D(\partial J_{-1}). $$ Indeed, $$ J_{-1}=J_a+J_{\tilde b}, $$ where $\tilde b(t)=b(t)+\frac{1}{2}t^2$ and we have $$ \partial J_a=-L_2,\quad\mbox{and}\quad \partial J_{\tilde b}=B+I. $$ In view of Proposition \ref{prop31} (a), we have $$ R(-L_2+B+2I)=E, $$ which implies that $-L_2+(B+I)$ is maximal monotone. From \cite[page 41]{Br-book} we have $$ \partial J_{-1}=\partial J_a+\partial J_{\tilde b}, $$ which proves the claim. Therefore $$ -L_2u_k+\beta(u_k)=(\nu_k-1)u_k,\quad k\in\mathbb{N}. $$ Set $\lambda_k=\nu_k-1$, $k\in\mathbb{N}$. By taking inner product with $u_k$ and taking into account that $\|u_k\|_E=R>0$ we have that $\lambda_k>0$. Finally, since $$ \varphi(u_k)\le\langle \partial\varphi(u_k),u_k\rangle=\nu_k R^2, $$ we have $\sup_{k\in\mathbb{N}}\lambda_k=\infty$. and thus \eqref{eigenprop} holds. \smallskip \noindent (i) In this part we shall use that $u\in W^{1,2}(H,\mu)$ implies that $|u|\in W^{1,2}(H,\mu)$, $J_a(|u|)=J_a(u)$, and moreover, since $\beta$ is odd, we also have $J_b(|u|)=J_b(u).$ We will apply Theorem 3 in \cite{Cl1}. To this end we set $$P:=\{v\in L^2(H,\mu) :\ v\ge 0\},\qquad I_P(u)=\begin{cases} 0\quad u\in P\\ +\infty\quad\mbox{otherwise},\end{cases}$$ and define $\varphi_+: E\to [0,\infty]$ by $\varphi_+(u)=\varphi(u)+I_P(u),$ $u\in E$. We have that $\varphi_+$ is convex, l.s.c., the set $\{u\in E:\varphi_+(u)\le\rho\}$ is compact for every $\rho\ge 0$, and $\varphi_+(0)=0$. We claim that $\overline{D(\varphi_+)}=P$. Indeed, let $u\in P$. By the density of $D(\varphi)$ in $E$, there exists $\{u_n\}\subseteq D(\varphi)$ such that $u_n\to u$ in $E$. Hence, $|u_n|\in D(\varphi_+)$ and $|u_n|\to |u|=u$ in $E$. Let $R>0$. From \cite[Theorem 3]{Cl1} there exists $(\nu,u)\in\mathbb{R}^+\times P$, with $\|u\|_E=R$, $\nu u\in D(\partial\varphi_+)$, $\nu u\in\partial\varphi_+(u)$ and $$ \varphi_+(u)=\inf_{v\in S_R}\varphi_+(v). $$ It follows that $$ \varphi_+(v)\ge\varphi_+(u)+\langle \nu u,v-u\rangle\quad\mbox{for all } v\in D(\varphi). $$ Since $u\in P$, we have $\varphi_+(u)=\varphi(u)$, hence $$ \varphi_+(v)\ge\varphi(u)+\langle \nu u,v-u\rangle\quad\mbox{for all } v\in D(\varphi). $$ Moreover, for all $v\in P\cap D(\varphi)$ we have $$ \varphi(v)\ge\varphi(u)+\langle \nu u,v-u\rangle, $$ hence for all $v\in D(\varphi)$ we have $$ \varphi(|v|)\ge\varphi(u)+\langle \nu u,|v|-u\rangle. $$ Since $\varphi(|v|)=\varphi(v)$, we obtain $$ \varphi(v)\ge\varphi(u)+\langle \nu u,v-u\rangle+\langle \nu u,|v|-v\rangle\ge\varphi(u) +\langle \nu u,v-u\rangle, $$ hence $\nu u\in D(\partial\varphi(u)$ and $\nu u=-L_2+Bu+u$. Setting now $\lambda=\nu-1$ we get $$ -L_2u+Bu=\lambda u. $$ Finally, we have \begin{align*} J_0(u)&=\varphi(u)-\frac{1}{2}R^2=\varphi_+(u)-\frac{1}{2}R^2\\ &=\inf_{v\in S_R}\varphi_+(v)-\frac{1}{2}R^2=\inf_{v\in S_R}\varphi(|v|)-\frac{1}{2}R^2\\ &=\inf_{v\in S_R}\varphi(v)-\frac{1}{2}R^2\\ &=\inf_{v\in S_R}J_0(v). \end{align*} \end{proof} We complete this note by exhibiting a class of functions $\beta$ for which the minimum of $J_0$ on $S_R$ is not attained at the constants for $R$ small. \begin{proposition}\label{prop42} Assume that $\beta$ satisfies the extra conditions \begin{gather} \lim_{s\to 0}\frac{b(s)}{s^2}=\infty,\quad\mbox{and}\quad \lim_{s\to \infty}\frac{b(s)}{s^2}=0;\label{mon}\\ \text{there exists $C>0$ such that $b(st)\le Cb(s)b(t)$ for all $s,t>0$.}\label{delta'} \end{gather} Then, there exists $R_0>0$ such that for any $R\in(0,R_0)$ $J_0$ does not achieve its minimum on $S_R$ at the constants. \end{proposition} \begin{proof} For $n\in\mathbb{N}$, we set $$ \tilde u_n(t)=\begin{cases} -n\alpha_n(|t|-\frac{1}{n}) & |t|\le \frac{1}{n}\\ 0 & |t|> \frac{1}{n} \end{cases} $$ where $\alpha_n$ will be chosen later. We define $u_n:H\to\mathbb{R}$ by $u_n(x):=\tilde u_n(\langle x,e_1\rangle)$ and we choose $\alpha_n$ so that $\|u_n\|_{ L^2(H,\mu)}=R$. We observe also that $u_n\in W^{1,2}(H,\mu)$. One verifies that \begin{equation}\label{ph3} C_1R\sqrt{n}\le\alpha_n\le C_2R\sqrt{n}, \end{equation} for some positive constants $C_1,\ C_2$. We will show now that if $n$ is chosen large enough and $R>0$ is small enough, then $$ J_0(u_n)0$ such that for any $R\in(0,R_0)$ $$K_0C_2^2n_0^2\frac{R^2}{b(R)}\le \frac{1}{4},$$ therefore from the first inequality in \eqref{ints} and \eqref{ph0}, we have \begin{equation}\label{ph5} \frac{1}{2}\int_H|Du_n|^2d\mu\le K_0C_2^2n_0^2\frac{R^2}{b(R)}b(R) \le \frac{1}{4}b(R). \end{equation} Hence, from \eqref{ph4} and \eqref{ph5} we conclude that for any $R\in(0,R_0)$, $$ \inf_{v\in S_R}J_0(v)\le J_0(u_{n_0})\le \frac{1}{2}b(R)=\frac{1}{2}J_0(R). $$ This completes the proof of the proposition. \end{proof} \begin{remark} \rm We note that $\beta(s)=|s|^{p-1}s$, $0