2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 207–222. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ASYMMETRIC ELLIPTIC PROBLEMS IN ℝ^{N}

JEAN-PIERRE GOSSEZ, LIAMIDI LEADI

Abstract. We work on the whole 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 Fučik spectrum.

1. Introduction

This work is mainly concerned with the following (asymmetric) eigenvalue problem

−Δ_{p}u = λ[m(x)(u^{+})^{p−1} − n(x)(u^{−})^{p−1}] in ℝ^{N}.  (1.1)

Here Δ_{p}u := div(|∇u|^{p−2}∇u), 1 < p < ∞, is the p-Laplacian, λ is a real parameter, m and n are weights whose properties will be specified later, and u^{±} = max{±u, 0}. Our assumptions on the weights will guarantee the existence of a unique positive principal eigenvalue λ_{1}(m) for the following (symmetric) eigenvalue problem

−Δ_{p}u = λm(x)|u|^{p−2}u in ℝ^{N}.  (1.2)

The principal motivation for considering problem (1.1) comes from the study of the Fučik spectrum. This spectrum is defined as the set Σ of those (α, β) ∈ ℝ^{2} such that

−Δ_{p}u = αm(x)(u^{+})^{p−1} − βn(x)(u^{−})^{p−1} in ℝ^{N}  (1.3)

has a nontrivial solution u. The relation between (1.3) and (1.1) is clear since the line of slope r through the origin of ℝ^{2} meets Σ at a point (α, β = rα) if and only if α is an eigenvalue of (1.1) for the weights m and rn.

Many works have been devoted to the study of the Fučik spectrum in the case of a bounded domain. But to our knowledge nothing has been done in the case of an unbounded domain, in particular ℝ^{N}, even in the linear case p = 2. It should be pointed out here that in ℝ^{N}, the presence of weights becomes essential since for instance (1.2) has no principal eigenvalue if m ≡ 1 ( cf. [4, 11] when N ≤ p, [15] when N > p).

2000 Mathematics Subject Classification. 35P30, 35J20, 35J60.

Key words and phrases. Principal eigenvalue; indefinite weight; unbounded domain;

p-Laplacian; Fučik spectrum.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

A study of (1.1) together with applications to the Fučik spectrum and to nonresonance was carried out recently in [2] in the case of a bounded domain Ω ⊂ ℝ^{N}:

−Δ_{p}u = λ[m(x)(u^{+})^{p−1} − n(x)(u^{−})^{p−1}] in Ω, u(x) = 0 on ∂Ω.  (1.4)

Denoting by μ_{1}(m) the first positive eigenvalue of the Dirichlet p-Laplacian with weight m on Ω and by ψ_{m} the associated normalized positive eigenfunction, it was shown in [2] that (1.4) always admits a positive nonprincipal eigenvalue, which in addition is the first eigenvalue of (1.4) greater than μ_{1}(m) and μ_{1}(n). This distinguished eigenvalue was constructed by applying a version of the mountain pass theorem to the functional ∫_{Ω} |∇u|^{p} restricted to the C^{1} manifold {u ∈ W_{0}^{1,p}(Ω) : ∫_{Ω}[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 ψ_{m} and −ψ_{n} were strict local minima.

When trying to adapt the above approach to the case of the whole ℝ^{N}, the relevant functional is

J(u) := ∫_{ℝ^{N}} |∇u|^{p}  (1.5)

restricted to

M_{m,n} := {u ∈ W : B_{m,n}(u) := ∫_{ℝ^{N}} [m(u^{+})^{p} + n(u^{−})^{p}] = 1},  (1.6)

where the space W is a suitable weighted Sobolev space on 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 3.3, whose proof uses some technique from [1, 11] as well as a result from [9] 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 3.1) as well as in the construction of some suitable auxiliary weights (cf. Lemma 4.2 and the proof of Theorem 4.1). One should also point out that the study of the continuous dependance of our distinguished eigenvalue of (1.1) 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 ℝ^{N} (cf. Proposition 4.7 and Corollary 4.8).

The existence of a positive nonprincipal eigenvalue for (1.1) 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 (1.1). We also study there some of the properties of c(m, n) as a function of (m, n). Section 5 is devoted to the Fučik spectrum. We show the existence of a first nontrivial curve in Σ ∩ (ℝ^{+} × 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 (1.2) and to various Sobolev imbeddings or Poincaré’s type inequalities to be used later.

2. 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:

(H_{1}) m_{1}, n_{1} ≥ 0, m_{1}, n_{1} ∈ L_{loc}^{∞}(ℝ^{N}) ∩ L^{s}(ℝ^{N}), where s = N/p if N > p and s = N_{0}/p for some integer N_{0} > p if N ≤ p; (H_{2}) m_{2}, n_{2} ≥ 0, m_{2}, n_{2} ∈ L_{loc}^{∞}(ℝ^{N}), with in addition m_{2}(x), n_{2}(x) ≥ ε_{0} for

some ε_{0} > 0 a.e. in ℝ^{N} if N ≤ p; (H_{3}) m^{+} ≢ 0, n^{+} ≢ 0; (H_{4}) for some a, b > 0: am_{2}(x) ≤ n_{2}(x) ≤ bm_{2}(x) a.e. in ℝ^{N}.

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}^{∞}(ℝ^{N}) with respect to the norm

∥u∥_{W} := [ ∫_{ℝ^{N}} (|∇u|^{p} + m_{2}|u|^{p})]^{1/p}.  (2.1)

Note that by (H_{4}), n_{2} would lead to the same space W. Note also that as observed in [9], the space W does not depend on the decomposition of m into m_{1} − m_{2}. The following imbeddings hold (cf. e.g. [3]): W ↪ D^{1,p}(ℝ^{N}) ↪ L^{p^{∗}}(ℝ^{N}) if N > p, W ↪ W^{1,p}(ℝ^{N}) ↪ L^{q}(ℝ^{N}) for all q ∈ [p, +∞[ if N = p and for all q ∈ [p, +∞] if N < p. Here D^{1,p}(ℝ^{N}) denotes when N > p the closure of C_{c}^{∞}(ℝ^{N}) with respect to the norm (∫_{ℝ^{N}} |∇u|^{p})^{1/p} and p^{∗} := Np/(N − p) is the critical Sobolev exponent. With s as in (H_{1}) above and s′ its Hölder conjugate, we will denote later by A the constant of the imbedding of D^{1,p}(ℝ^{N}) into L^{ps′}(ℝ^{N}) = L^{p^{∗}}(ℝ^{N}) when N > p, and by B the constant of the imbedding of W^{1,p}(ℝ^{N}) into L^{ps′}(ℝ^{N}) when N ≤ p. One also has the compact imbedding of W into L^{p}(m_{1}, ℝ^{N}), the L^{p} space on ℝ^{N} with weight m_{1} (cf. [9]).

By a solution u of (1.1) (or of related equations), we mean a weak solution, i.e. u ∈ W with

∫_{ℝ^{N}} |∇u|^{p−2}∇u∇v = λ ∫_{ℝ^{N}} [m(u^{+})^{p−1} − n(u^{−})^{p−1}]v ∀v ∈ W.  (2.2)

Note that by the above imbeddings, every integral in (2.2) is well-defined. Regularity results from [14] and [16] on general quasilinear equations imply that such a weak solution u belongs to C^{1}(ℝ^{N}). It is also known that if N < p, or if N > p and (H_{4}) is replaced by (H′_{4}) (cf. Remark 2.5 below), then a weak solution u decays to zero at infinity (cf. [3] for N < p, [8, 10] for N > p).

Let us define

λ_{1}(m) := inf { ∫_{ℝ^{N}} |∇u|^{p} : u ∈ W and ∫_{ℝ^{N}} m|u|^{p} = 1}.

It is known (cf. [1, 7, 8, 9, 10]) that this infimum is achieved and that λ_{1}(m) is the unique positive principal eigenvalue of (1.2). (By a principal eigenvalue, we mean an eigenvalue associated to an eigenfunction which does not change sign). Moreover λ_{1}(m) is simple and admits an eigenfunction φ_{m} ∈ W ∩ C^{1}(ℝ^{N}), with φ_{m}(x) > 0 in ℝ^{N} and ∫_{ℝ^{N}} mφ_{m}^{p} = 1. One also knows that λ_{1}(m) is isolated in the spectrum, which implies

λ_{2}(m) := inf {λ ∈ ℝ : λ eigenvalue of (1.2) with λ > λ_{1}(m)} > λ_{1}(m). (2.3)

As we will see later (cf. Remark 3.4 or Theorem 4.1), this infimum (2.3) is also achieved, and consequently λ_{2}(m) is really the second positive eigenvalue of (1.2).

Remark 2.1. Assume that m satisfies (H_{1}), (H_{2}), (H_{3}) and that m^{−} ≢ 0. If N > p and m_{2} ∈ L^{N/p}(ℝ^{N}), then (1.2) also has a unique negative principal eigenvalue λ_{−1}(m) = −λ_{1}(−m). If N ≤ p, then (1.2) generally does not admit a negative principal eigenvalue; this follows from the nonexistence results of [4] for p = 2 and [11] for p ≠ 2.

The following lemma will play a role in low dimensions. It can be easily derived from the proof in [1, Theorem 3]. (The assumptions about N_{0} in (H_{1}) and about ε_{0} in (H_{2}) are used here).

Lemma 2.2. Let N ≤ p. There exists C = C(m_{1}, m_{2}, n_{1}, n_{2}, N, p, N_{0}) such that

∫_{ℝ^{N}} |u|^{p} ≤ C ∫_{ℝ^{N}} |∇u|^{p}  (2.4)

for all u ∈ W satisfying B_{m,n}(u) ≥ 0. Moreover the constant C in (2.4) can be chosen so as to remain bounded when m_{1} and n_{1} vary in a bounded subset of L^{N_{0}/p}(ℝ^{N}).

Remark 2.3. 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) ≥ some ε_{0} and am_{2}(x) ≤ n_{2}(x) ≤ bn_{2}(x) are assumed to hold only for a.e. x with |x| ≥ R for some R ≥ 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 = m̃_{1} − m̃_{2}, n = ñ_{1} − ñ_{2} where m̃_{1}, m̃_{2}, ñ_{1}, ñ_{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 = m̃_{1} − m̃_{2}, n = ñ_{1} − ñ_{2} now satisfy (H_{1}), (H_{2}), (H_{3}), (H_{4}).

Remark 2.4. In the situation of Remark 2.3, one can also show that the space W associated to m_{2} coincides with the space W associated to m̃_{2}. (The proof of this fact uses the inequality that if Ω is a smooth bounded domain and if E is a subset of Ω of positive measure, then there exists a constant c such that ∥u∥_{L^{p}(Ω)} ≤ c(∥u∥_{L^{p}(E)} + ∥∇u∥_{L^{p}(Ω)}) for all u ∈ W^{1,p}(Ω)). 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}.

Remark 2.5. In high dimensions, assumption (H_{4}) can be replaced in all this work by

(H′_{4}) N > p and m_{2}, n_{2} ∈ L^{N/p}(ℝ^{N}). This situation is in fact much simpler (for instance W is then equal to D^{1,p}(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 (1.1) and (1.3).

Let us conclude this section with some general definitions relative to the (PS) condition. Let E be a real Banach space and

M := {u ∈ E : g(u) = 1},  (2.5)

where g ∈ C^{1}(E, ℝ) and 1 is a regular value of g. Let f ∈ C^{1}(E, ℝ) and denote by f̃ the restriction of f to M. The differential of f̃ at u ∈ M has a norm which will be denoted by ∥f̃′(u)∥_{∗} and which is given by the norm of the restriction of f′(u) ∈ E^{∗} to the tangent space of M at u:

T_{u}M := {v ∈ E : ⟨g′(u), v⟩ = 0},

where ⟨, ⟩ denotes the pairing between E and its dual E^{∗}. A critical point of f̃ is a point u ∈ M such that ∥f̃′(u)∥_{∗} = 0; f̃(u) is then called a critical value of f̃. We recall that f̃ is said to satisfy the (PS) condition if for any sequence u_{k} ∈ M such that f̃(u_{k}) is bounded and ∥f̃′(u_{k})∥_{∗} → 0, one has that u_{k} admits a convergent subsequence.

3. Construction of a nonprincipal eigenvalue

In this section and in the following one, we consider the eigenvalue problem (1.1). 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 λ of (1.1) with λ > 0. Clearly the only positive principal eigenvalues of (1.1) are λ_{1}(m) and λ_{1}(n). Moreover multiplying by u^{+} or u^{−}, one easily sees that if (1.1) with λ > 0 has a solution which changes sign, then λ > max{λ_{1}(m), λ_{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 (1.5) and (1.6), which are C^{1} functionals on W, and the restriction J̃ of J to the manifold M_{m,n} defined in (1.6). In this context one easily verifies that λ > 0 is an eigenvalue of (1.1) if and only if λ is a critical value of J̃.

A first critical point of J̃ comes from global minimization. Indeed

J̃(u) ≥ λ_{1}(m)[ ∫_{ℝ^{N}} m(u^{+})^{p}]^{+} + λ_{1}(n)[ ∫_{ℝ^{N}} n(u^{−})^{p}]^{+} ≥ min{λ_{1}(m), λ_{1}(n)}

for all u ∈ M_{m,n}, and one has J̃(u) = min{λ_{1}(m), λ_{1}(n)} for either u = φ_{m} or u = −φ_{n}. Consequently either φ_{m} or −φ_{n} is a global minimum of J̃ and so a critical point of J̃.

A second critical point of J̃ comes from the following proposition.

Proposition 3.1. φ_{m} and −φ_{n} are strict local minimum of J̃, with corresponding critical values λ_{1}(m) and λ_{1}(n).

Proof. The present proof is partly different from that of the analogous result in [2]; the difficulty lies at the level of [2, Lemma 3]. We adapt to our situation some technique from [7].

Let us consider φ_{m} (similar argument for −φ_{n}). Assume by contradiction the existence of a sequence u_{k} ∈ M_{m,n} with u_{k} ≠ φ_{m}, u_{k} → φ_{m} in W and J̃(u_{k}) ≤ λ_{1}(m). We first observe that u_{k} changes sign for k sufficiently large. Indeed, since u_{k} → φ_{m}, u_{k} must be > 0 somewhere. If u_{k} ≥ 0 in ℝ^{N}, then

J̃(u_{k}) = ∫_{ℝ^{N}} |∇u_{k}|^{p} > λ_{1}(m) ∫_{ℝ^{N}} mu_{k}^{p} = λ_{1}(m)

since u_{k} ≠ φ_{m} and u_{k} ∈ M_{m,n}. But this contradicts J̃(u_{k}) ≤ λ_{1}(m). So u_{k} changes sign for k sufficiently large. Now we have

λ_{1}(m) ∫_{ℝ^{N}} [m(u_{k}^{+})^{p} + n(u_{k}^{−})^{p}] = λ_{1}(m) ≥ J̃(u_{k})
= ∫_{ℝ^{N}} |∇u_{k}^{+}|^{p} + ∫_{ℝ^{N}} |∇u_{k}^{−}|^{p}
≥ λ_{1}(m) ∫_{ℝ^{N}} m(u_{k}^{+})^{p} + ∫_{ℝ^{N}} |∇u_{k}^{−}|^{p}.

Consequently

λ_{1}(m) ∫_{ℝ^{N}} n(u_{k}^{−})^{p} ≥ ∫_{ℝ^{N}} |∇u_{k}^{−}|^{p}.  (3.1)

Let v_{k} := u_{k}^{−}/(∫_{ℝ^{N}} |∇u_{k}^{−}|^{p})^{1/p} and Ω_{k}^{−} := {x ∈ ℝ^{N} : u_{k}(x) < 0}. We deduce from (3.1) that

\frac{1}{λ_{1}(m)} ≤ ∫_{ℝ^{N}} n(v_{k})^{p} ≤ ∫_{Ω_{k}^{−}} n_{1}(v_{k})^{p}.  (3.2)

Consider first the case N > p. We deduce from (3.2), using Hölder inequality, that

\frac{1}{λ_{1}(m)} ≤ ∥n_{1}∥_{L^{s}(Ω_{k}^{−})}∥v_{k}∥_{L^{p^{∗}} (ℝ^{N})}^{p} ≤ A^{p}∥n_{1}∥_{L^{s}(Ω_{k}^{−})},

where the imbedding constant A was defined in Section 2, and consequently

∥n_{1}∥_{L^{s}(Ω_{k}^{−})} ≥ \frac{1}{A^{p}λ_{1}(m)} = ε.  (3.3)

Take now r > 0 sufficiently large so that ∥n_{1}∥_{L^{s}(B_{r}^{c})}^{s} ≤ ε^{s}/2, where B_{r}^{c} = ℝ^{N}\B_{r} and B_{r} denotes the ball of radius r centred at the origin. We deduce from (3.3) that ∥n_{1}∥_{L^{s}(Ω_{k}^{−}∩B_{r})}^{s} ≥ ε^{s}/2, and consequently

|Ω_{k}^{−} ∩ B_{r}| ≥ \frac{ε^{s}}{2∥n_{1}∥_{L^{∞}(B_{r})}^{s}} > 0,  (3.4)

where |E| denotes the measure of the set E. Since u_{k} → φ_{m} in L^{p}(B_{r}) and φ_{m}(x) > 0 for all x ∈ B_{r}, one has that |{x ∈ B_{r} : u_{k}(x) < 0}| → 0. But this contradicts (3.4).

In the case N ≤ p, we have a similar situation. Indeed using Hölder inequality, the imbedding of W^{1,p}(ℝ^{N}) into L^{ps′}(ℝ^{N}) (with constant B, cf. Section 2) and Lemma 2.2 (with constant C), one derives from (3.2) that

\frac{1}{λ_{1}(m)} ≤ (1 + C)B^{p}∥n_{1}∥_{L^{s}(Ω_{k}^{−})}.  (3.5)

The conclusion then follows as in the case N > p. □

To get a third critical point of J̃, 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}:

Γ := {γ ∈ C([−1, 1], M_{m,n}) : γ(−1) = φ_{m} and γ(1) = −φ_{n}}.  (3.6)

Arguing as in [2, p. 589 ], one shows that Γ is nonempty, and so the minimax value

c(m, n) := inf_{γ∈Γ} max_{u∈γ([−1,1])} J̃(u),  (3.7)

is finite. The following is the main result in this section.

Theorem 3.2. c(m, n) is an eigenvalue of (1.1) which satisfies

max{λ_{1}(m), λ_{1}(n)} < c(m, n).  (3.8)

The rest of this section is devoted to the proof of the above theorem. We first consider the (PS) condition.

Proposition 3.3. The functional J̃ satisfies the (PS) condition on M_{m,n} Proof. Let u_{k} ∈ M_{m,n} be a (PS) sequence for J̃. So ∫_{ℝ^{N}} |∇u_{k}|^{p} remains bounded and for some ε_{k} → 0,

| ∫_{ℝ^{N}} |∇u_{k}|^{p−2}∇u_{k}∇w| ≤ ε_{k}∥w∥_{W}  (3.9)

for all w ∈ 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}(ℝ^{N}) and consequently in L^{p^{∗}}(ℝ^{N}). Using B_{m,n}(u_{k}) = 1 and (H_{4}), one has

min(a, 1) ∫_{ℝ^{N}} m_{2}|u_{k}|^{p} ≤ −1 + ∫_{ℝ^{N}} [m_{1}(u_{k}^{+})^{p} + n_{1}(u_{k}^{−})^{p}],  (3.10)

where the right hand side remains bounded (by (H_{1}) and Hölder inequality). Consequently ∫_{ℝ^{N}} m_{2}|u_{k}|^{p} remains bounded. In the case N ≤ p, Lemma 2.2 implies that u_{k} remains bounded in W^{1,p}(ℝ^{N}) and consequently in L^{ps′}(ℝ^{N}). One then again deduces from (3.10) that ∫_{ℝ^{N}} m_{2}|u_{k}|^{p} remains bounded. Hence in any case N > p or N ≤ p, u_{k} remains bounded in W. It follows that for a subsequence (still denoted by u_{k}), u_{k} → u weakly in W, strongly in L^{p}(m_{1}, ℝ^{N}) and in L^{p}(n_{1}, ℝ^{N}), and ∫_{ℝ^{N}} |∇u_{k}|^{p} converges.

In the rest of the proof we will assume N > p (a similar argument holds in the case N ≤ p). Observe that if w ∈ W, then (w − a_{k}(w)u_{k}) ∈ T_{u_{k}}(M_{m,n}) where a_{k}(w) := ∫_{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 (3.9), one deduces

∫_{ℝ^{N}} |∇u_{k}|^{p−2}∇u_{k}∇(u_{k} − u_{l}) = t_{k} ∫_{ℝ^{N}} [m(u_{k}^{+})^{p−1} − n(u_{k}^{−})^{p−1}](u_{k} − u_{l}) + 0(ε_{k}),

where t_{k} := ∫_{ℝ^{N}} |∇u_{k}|^{p}. This implies

\begin{aligned}0 & ≤ ∫_{ℝ^{N}} (|∇u_{k}|^{p−2}∇u_{k} − |∇u_{l}|^{p−2}∇u_{l})∇(u_{k} − u_{l}) \\ & = t_{k} ∫_{ℝ^{N}} m[(u_{k}^{+})^{p−1} − (u_{l}^{+})^{p−1}](u_{k} − u_{l}) \\ & + t_{k} ∫_{ℝ^{N}} n[−(u_{k}^{−})^{p−1} + (u_{l}^{−})^{p−1}](u_{k} − u_{l}) \\ & + (t_{k} − t_{l}) ∫_{ℝ^{N}} [m(u_{l}^{+})^{p−1} − n(u_{l}^{−})^{p−1}](u_{k} − u_{l}) + 0(ε_{k}) + 0(ε_{l}) \\ & ≤ t_{k}(I_{1} + I_{2}) + |t_{k} − t_{l}|I_{3} + 0(ε_{k}) + 0(ε_{l}), \end{aligned}  (3.11)

where

I_{1} := ∫_{ℝ^{N}} m_{1}[(u_{k}^{+})^{p−1} − (u_{l}^{+})^{p−1}](u_{k} − u_{l}),
I_{2} := ∫_{ℝ^{N}} n_{1}[−(u_{k}^{−})^{p−1} + (u_{l}^{−})^{p−1}](u_{k} − u_{l}),
I_{3} := ∫_{ℝ^{N}} |m(u_{l}^{+})^{p−1} − n(u_{l}^{−})^{p−1}∥u_{k} − u_{l}|.

We claim that the right hand side of (3.11) approaches zero when k, l → +∞. Indeed using Hölder inequality and the strong convergence of u_{k} in L^{p}(m_{1}, ℝ^{N}), one sees that I_{1} → 0. Similarly I_{2} → 0 . Furthermore Hölder inequality implies that I_{3} remains bounded. Since (t_{k} − t_{l}) → 0, we conclude that the right hand side of (3.11) goes to 0 as k, l → +∞, and the claim is proved .

To go on in the proof of Proposition 3.3, we observe that for some constant d = d(p) and for any α, β ∈ ℝ^{N},

|α − β|^{p} ≤ d{(|α|^{p−2}α − |β|^{p−2}β)(α − β)}^{r/2}(|α|^{p} + |β|^{p})^{1−r/2},  (3.12)

where r = p if p ∈]1, 2[ and r = 2 if p ≥ 2 (cf. [13]). Applying (3.12) and Hölder inequality, one easily derives from the claim that ∇u_{k} → ∇u in L^{p}(ℝ^{N}). Moreover the calculation leading to (3.11) gives, using (H_{4}),

0 ≤ min(a, 1)t_{k} ∫_{ℝ^{N}} m_{2}(|u_{k}|^{p−2}u_{k} − |u_{l}|^{p−2}u_{l})(u_{k} − u_{l})
≤ t_{k}(I_{1} + I_{2}) + |t_{k} − t_{l}|I_{3} + 0(ε_{k}) + 0(ε_{l}),

where the right hand side goes to zero by the claim. We then deduce from the above that ∫_{ℝ^{N}} m_{2}|u_{k} − u_{l}|^{p} → 0 by applying successively (3.12), Hölder inequality and the fact that lim_{k→+∞} t_{k} = ∫_{ℝ^{N}} |∇u|^{p} ≠ 0 (the latter quantity is nonzero because u ∈ W and W does not contain any nonzero constant). Consequently u_{k} → u in W and Proposition 3.3 is proved. □

Remark 3.4. The arguments in the above proof can be used to show that the positive part of the spectrum of (1.2) is closed (cf. chap.2 of [12] for details).

We now have all the ingredients for the next proof.

Proof of Theorem 3.2. The conclusion follows by applying the mountain pass theorem on a C^{1} manifold as given in [2, Proposition 4] or in [5, Proposition 2.1]: the (PS) condition is provided by Proposition 3.3 and the geometry comes by combining Proposition 3.1 with [2, Lemma 6]. □

4. A first nonprincipal eigenvalue

We have seen at the beginning of Section 3 that λ_{1}(m) and λ_{1}(n) are the first two positive eigenvalues of (1.1). The present section is mainly devoted to the proof that the eigenvalue c(m, n) constructed in (3.7) is the next positive eigenvalue of (1.1).

Theorem 4.1. Problem (1.1) does not admit any eigenvalue between the values max{λ_{1}(m), λ_{1}(n)} and c(m, n).

Proof. The present proof is partly different from that of the analogous result in [2]; the difficulty lies at the level of the construction of some auxiliary weights.

Assume by contradiction that there exists an eigenvalue λ of problem (1.1) with max{λ_{1}(m), λ_{1}(n)} < λ < c(m, n). Our goal is to construct a path in Γ on which J̃ remains ≤ λ, which yields a contradiction with the definition (3.7) of c(m, n).

Let u ∈ M_{m,n} be a critical point of J̃ at level λ. Since u changes sign, one obtains from the equation satisfied by u,

0 < ∫_{ℝ^{N}} |∇u^{+}|^{p} = λ ∫_{ℝ^{N}} m(u^{+})^{p} and 0 < ∫_{ℝ^{N}} |∇u^{−}|^{p} = λ ∫_{ℝ^{N}} n(u^{−})^{p}. (4.1)

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} ≡ (u^{+})_{m,n} by writing

γ_{1}(t) := [tu + (1 − t)u^{+}]_{m,n} , t ∈ [0, 1].  (4.2)

Using (4.1), it is easy to show that γ_{1}(t) is well-defined, belongs to M_{m,n} and satisfies J̃(γ_{1}(t)) = λ ∀t ∈ [0, 1]. In a similar way we go from u to (−u^{−})_{m,n} in M_{m,n} by staying at level λ. We now describe the construction of a path in M_{m,n} from v to φ_{m} which stays at levels ≤ λ. A similar construction would yield a path in M_{m,n} from (−u^{−})_{m,n} to −φ_{n} which stays at levels ≤ λ. Putting everything together, we get the desired path from φ_{m} to −φ_{n}.

To construct the path from v to φ_{m}, first consider the manifold M_{m,m}. Clearly v ∈ M_{m,m}. The critical points of the restriction of J to M_{m,m} are the normalized eigenfunctions of (1.2). 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 ν :] − ϵ, ϵ[→ M_{m,m} with ν(0) = v and \frac{d}{dt}J(ν(t))|_{t=0} ≠ 0. Following a little portion of this path ν in the positive or negative direction ( call ν_{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) = λ, lies at levels < λ. The path γ_{2}(t) = |ν_{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) = λ, at levels < λ (since J(|ν_{1}(t)|) = J(ν_{1}(t)) ).

Let now m_{(ϵ)} be defined for 0 ≤ ϵ ≤ 1 by

m_{(ϵ)} = \begin{cases}m & if m < 0 \\ ϵm_{1} − m_{2} & if m ≥ 0. \end{cases}

One has m_{(1)} ≡ m, m_{(1)}^{+} ≢ 0, m_{(0)} ≤ 0 and so m_{(0)}^{+} ≡ 0. Hence there exists ϵ_{0} ∈]0, 1[ such that

m_{(ϵ)}^{+} ≢ 0 if ϵ_{0} < ϵ < 1,
m_{(ϵ)}^{+} ≡ 0 if 0 ≤ ϵ ≤ ϵ_{0}.

Using Lemma 4.2 below, we see that for ϵ > ϵ_{0} close to ϵ_{0}, m_{(ϵ)} 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 λ_{1}(l) > λ and l ≤ m in ℝ^{N}. Fix such an ϵ. We then consider the manifold M_{m,l} and the sublevel set

O := {u ∈ M_{m,l} : J(u) < λ}.

Clearly v_{1} and φ_{m} ∈ O (because they belong to M_{m,m}, are ≥ 0 and have the right levels). Moreover the only critical point in O of the restriction of J to M_{m,l} is φ_{m} (because the first two critical levels λ_{1}(m) and λ_{1}(l) verify λ_{1}(m) < λ < λ_{1}(l)). Applying [2, Lemma 14] to the component of O which contains v_{1}, we get a path γ_{3} in O from v_{1} to φ_{m}. We then consider the path

γ_{4}(t) := \frac{|γ_{3}(t)|}{(∫_{ℝ^{N}} m|γ_{3}(t)|^{p})^{1/p}} .

By the choice of l, one has

1 = ∫_{ℝ^{N}} [m(γ_{3}(t)^{+})^{p}+l(γ_{3}(t)^{−})^{p}] ≤ ∫_{ℝ^{N}} [m(γ_{3}(t)^{+})^{p}+m(γ_{3}(t)^{−})^{p}] = ∫_{ℝ^{N}} m|γ_{3}(t)|^{p},

and consequently γ_{4} is well-defined. Moreover γ_{4} goes from v_{1} to φ_{m} and belongs to M_{m,n}. Finally

J(γ_{4}(t)) = \frac{∫_{ℝ^{N}} |∇γ_{3}(t)|^{p}}{∫_{ℝ^{N}} m|γ_{3}(t)|^{p}} ≤ ∫_{ℝ^{N}} |∇γ_{3}(t)|^{p} < λ,

since γ_{3}(t) ∈ O. The path γ_{4} thus allows us to move from v_{1} to φ_{m} in M_{m,n} by staying at levels < λ. □

Lemma 4.2. λ_{1}[m_{(ϵ)}] → ∞ as ϵ ↓ ϵ_{0}.

Proof. One has

\frac{1}{λ_{1}[m_{(ϵ)}]} = \frac{∫_{ℝ^{N}} m_{(ϵ)}φ_{m_{(ϵ)}}^{p}}{∫_{ℝ^{N}} |∇φ_{m_{(ϵ)}}|^{p}} ≤ \frac{∥(ϵm_{1} − m_{2})^{+}∥_{L^{s}(ℝ^{N})}∥φ_{m_{(ϵ)}}∥_{L^{ps′} (ℝ^{N})}^{p}}{∫_{ℝ^{N}} |∇φ_{m_{(ϵ)}}|^{p}} .  (4.3)

If N > p, one deduces from (4.3) that 1/λ_{1}[m_{(ϵ)}] ≤ A∥(ϵm_{1} − m_{2})^{+}∥_{L^{s}(ℝ^{N})}, and the conclusion follows since (ϵm_{1} − m_{2})^{+} ∈ L^{s}(ℝ^{N}) and (ϵm_{1} − m_{2})^{+} ↓ 0 as ϵ ↓ ϵ_{0}. If N ≤ p, since B_{m_{(ϵ)},n_{(ϵ)}}(φ_{m_{(ϵ)}}) = 1, one has

∥φ_{m_{(ϵ)}}∥_{L^{ps′} (ℝ^{N})}^{p} ≤ B^{p}(1 + C(ϵ)) ∫_{ℝ^{N}} |∇φ_{m_{(ϵ)}}|^{p},

where C(ϵ) = C(m_{(ϵ)}, N, p, N_{0}) is the constant from Lemma 2.2 (which remains bounded as ϵ varies in ]ϵ_{0}, 1[) . The conclusion then follows from (4.3) as in the case N > p. □

Theorem 4.1 for m ≡ n yields the following variational characterization of the second eigenvalue of the p-Laplacian with weight on ℝ^{N} (cf. (2.3)).

Corollary 4.3. One has

λ_{2}(m) = inf_{γ∈Γ_{0}} max_{u∈γ([−1,1])} ∫_{ℝ^{N}} |∇u|^{p},

where Γ_{0} := {γ ∈ C([−1, 1], M_{m,m}) : γ(−1) = φ_{m} and γ(1) = −φ_{m}} and M_{m,m} := {u ∈ W : ∫_{ℝ^{N}} m|u|^{p} = 1}.

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 [2].

Proposition 4.4. One has

c(m, n) = inf_{γ∈Γ_{1}} max_{t∈[−1,1]} J(γ(t)),  (4.4)

where Γ_{1} := {γ ∈ C([−1, 1], M_{m,n}) : γ(−1) ≥ 0 and γ(1) ≤ 0}.

Proposition 4.5. Let m = m_{1} − m_{2}, n = n_{1} − n_{2}, m̂ = m̂_{1} − m̂_{2}, n̂ = n̂_{1} − n̂_{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 m̂, n̂. If m_{1} ≤ m̂_{1}, n_{1} ≤ n̂_{1}, m̂_{2} ≤ m_{2} and n̂_{2} ≤ n_{2} a.e. in ℝ^{N}, then c(m, n) ≥ c(m̂, n̂). If in addition

∫_{ℝ^{N}} (m̂ − m)(u^{+})^{p} + ∫_{ℝ^{N}} (n̂ − n)(u^{−})^{p} > 0

for at least one eigenfunction u associated to c(m, n), then c(m, n) > c(m̂, n̂).

Proof. Denote by W (resp. Ŵ) the weighted Sobolev space defined in Section 2 and associated to the weight m_{2} (resp. m̂_{2}). One clearly has W ⊂ Ŵ. Once this has been observed, the proof is easily adapted from that of [2, Propositions 23 and 25]. One uses in particular the fact that if the path γ is admissible in formula (3.7) for c(m, n), then γ(t) ∈ Ŵ for all t ∈ [−1, 1], and so by normalization one can construct a path in Ŵ which will be admissible in formula (4.4) for c(m̂, n̂). □

Let us also observe that definition (3.7) clearly implies that c(m, n) is homogeneous of degree −1: c(sm, sn) = c(m, n)/s for s > 0. Some sort of separate sub-homogeneity also holds, which will be useful later:

Proposition 4.6. Assume that the weights m, n satisfy hypothesis (H_{1}), (H_{2}), (H_{3}) and (H_{4}). If 0 < s < ŝ, then c(ŝm, n) < c(sm, n) and c(m, ŝn) < c(m, sn).

Proof. Let us first observe that the pair of weights ŝm, n (as well as the other pairs of weights appearing above) also satisfy hypothesis (H_{1}), (H_{2}), (H_{3}) and (H_{4}). Moreover the “m_{2} parts” of all these weights are comparable in the sense of hypothesis (H_{4}), which implies that a single weighted Sobolev space W can be used. Once this has been observed, the proof can be easily adapted from [2, Proposition 31]. □

We finally turn to the study of the continuous dependance of c(m, n) with respect to m, n. The situation here is more involved than in [2].

Proposition 4.7. Let m = m_{1}−m_{2}, n = n_{1}−n_{2} satisfy (H_{1}), (H_{2}), (H_{3}) and (H_{4}), and let m_{k} = m_{1k} − m_{2k}, n_{k} = n_{1k} − n_{2k} with m_{1k}, m_{2k}, n_{1k}, n_{2k} ≥ 0, k = 1, 2, . . .. Assume that m_{1k}, n_{1k} belong to L_{loc}^{∞}(ℝ^{N})∩L^{s}(ℝ^{N}) and converge in L^{s}(ℝ^{N}) to m_{1}, n_{1} respectively. Assume that m_{2k}, n_{2k} belong to L_{loc}^{∞}(ℝ^{N}) and converge to m_{2}, n_{2} respectively in the following sense: for some ε_{k} → 0,

|m_{2k} − m_{2}| ≤ ε_{k}m_{2} and |n_{2k} − n_{2}| ≤ ε_{k}n_{2} a.e. in ℝ^{N}.  (4.5)

Then c(m_{k}, n_{k}) → c(m, n).

The convergence (4.5) is unusual but Proposition 4.7 will suffice to derive later the continuity of the first curve in the Fučik spectrum.

Proof of Proposition 4.7. Observe that by our assumptions, the “m_{2} parts” of all the weights m, n, m_{k}, n_{k} are comparable in the sense of hypothesis (H_{4}) (with constants a, b independent of k), and so a single space W is involved.

We first prove the upper semicontinuity. Let ε > 0 and take γ ∈ Γ such that max_{t} J(γ(t)) < c(m, n) + ε. Let γ_{k}(t) := γ(t)/B_{m_{k},n_{k}}(γ(t))^{1/p}. We will show that γ_{k} is well defined and that

max_{t} J(γ_{k}(t)) < c(m, n) + ε  (4.6)

for k sufficiently large. Once this is done, one deduces from Proposition 4.4 that c(m_{k}, n_{k}) < c(m, n) + ε and consequently, since ε > 0 is arbitrary, that lim sup c(m_{k}, n_{k}) ≤ c(m, n).

The path γ_{k} is clearly well defined if B_{m_{k},n_{k}}(γ(t)) > 0 ∀t ∈ [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}}(γ(t_{k})) ≤ 0 for some t_{k} ∈ [0, 1]. For a further subsequence, t_{k} → t_{0} and γ(t_{k}) → γ(t_{0}) in W and a.e. in ℝ^{N}. We claim that

B_{m,n}(γ(t_{0})) ≤ 0,  (4.7)

which is impossible since γ ∈ Γ. Deriving (4.7) is a matter of going to the limit in the expression B_{m_{k},n_{k}}(γ(t_{k})) ≤ 0. The “m_{1k} (or n_{1k}) terms” can be handled by Hölder inequality in a standard way. To handle the “m_{2k} (or n_{2k}) terms”, we observe that γ(t_{k}) → γ(t_{0}) in L^{p}(m_{2}, ℝ^{N}), which means that m_{2}^{1/p}γ(t_{k}) → m_{2}^{1/p}γ(t_{0}) in L^{p}(ℝ^{N}), and consequently, for a further subsequence and for some v ∈ L^{p}(ℝ^{N}), |m_{2}^{1/p}γ(t_{k})| ≤ v a.e. in ℝ^{N}. This inequality and the fact that m_{2k} and n_{2k} are controlled by m_{2} (a consequence of (H_{4}) and (4.5)) allow the use of the dominated convergence theorem to handle these “m_{2k} (or n_{2k}) terms”. Let us now prove that (4.6) holds. Write max_{t} J(γ_{k}(t)) = J(γ_{k}(τ_{k})) and assume by contradiction that for a subsequence

J(γ_{k}(τ_{k})) ≥ c(m, n) + ε.  (4.8)

The preceding argument shows that for a further subsequence, one has τ_{k} → τ_{0} and B_{m_{k},n_{k}}(γ(τ_{k})) → B_{m,n}(γ(τ_{0})) = 1. Consequently, by (4.8), J(γ(τ_{0})) ≥ c(m, n) + ε, a contradiction with the choice of γ.

To prove the lower semicontinuity, suppose by contradiction that for a subsequence, one has c(m_{k}, n_{k}) → c_{0} with c_{0} < c(m, n). Let u_{k} ∈ 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 ∫_{ℝ^{N}} |∇u_{k}|^{p} = c(m_{k}, n_{k}) and so ∫_{ℝ^{N}} |∇u_{k}|^{p} remains bounded. To get a bound on ∫_{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 2.2 when N ≤ 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 (4.5), one deduces the desired bound on ∫_{ℝ^{N}} m_{2}|u_{k}|^{p}. It follows that for a subsequence, u_{k} → u weakly in W, strongly in L^{p}(m_{1}, ℝ^{N}) and in L^{p}(n_{1}, ℝ^{N}).

We will now prove that u_{k} → 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

∫_{ℝ^{N}} |∇u_{k}|^{p−2}∇u_{k}∇(u_{k} − u_{l}) = c_{k} ∫_{ℝ^{N}} [m_{k}(u_{k}^{+})^{p−1} − n_{k}(u_{k}^{−})^{p−1}](u_{k} − u_{l}),

which implies

0 ≤ ∫_{ℝ^{N}} (|∇u_{k}|^{p−2}∇u_{k} − |∇u_{l}|^{p−2}∇u_{l})∇(u_{k} − u_{l})
= c_{k} ∫_{ℝ^{N}} m_{k}[(u_{k}^{+})^{p−1} − (u_{l}^{+})^{p−1}](u_{k} − u_{l})
+ c_{k} ∫_{ℝ^{N}} n_{k}[−(u_{k}^{−})^{p−1} + (u_{l}^{−})^{p−1}](u_{k} − u_{l})
+ (c_{k} − c_{l}) ∫_{ℝ^{N}} [m_{k}(u_{l}^{+})^{p−1} − n_{k}(u_{l}^{−})^{p−1}](u_{k} − u_{l})
+ c_{l} ∫_{ℝ^{N}} [(m_{k} − m_{l})(u_{l}^{+})^{p−1} − (n_{k} − n_{l})(u_{l}^{−})^{p−1}](u_{k} − u_{l})
≤ c_{k}(I_{1} + I_{2}) + |c_{k} − c_{l}|I_{3} + c_{l}I_{4},

where

I_{1} = ∫_{ℝ^{N}} m_{1k}[(u_{k}^{+})^{p−1} − (u_{l}^{+})^{p−1}](u_{k} − u_{l}),
I_{2} = ∫_{ℝ^{N}} n_{1k}[−(u_{k}^{−})^{p−1} + (u_{l}^{−})^{p−1}](u_{k} − u_{l}),
I_{3} = ∫_{ℝ^{N}} [|m_{k}|(u_{k}^{+})^{p−1} + |n_{k}|(u_{l}^{−})^{p−1}]|u_{k} − u_{l}|,
I_{4} = ∫_{ℝ^{N}} [|m_{k} − m_{l}|(u_{l}^{+})^{p−1} + |n_{k} − n_{l}|(u_{l}^{−})^{p−1}]|u_{k} − u_{l}|.

Arguing as in the proof of Proposition 3.3 , one then easily proves that u_{k} → u in W. (The full strength of the convergence (4.5) is used here to verify that I_{4} → 0).

The limit u ∈ M_{m,n} and is a solution of (1.1) for λ = c_{0}. Since c_{0} < c(m, n), Theorem 4.1 implies that c_{0} = λ_{1}(m) and u = φ_{m}, or c_{0} = λ_{1}(n) and u = −φ_{n}. Consider the first case (similar argument in the second one). Writing v_{k} = u_{k}^{−}/(∫_{ℝ^{N}} |∇u_{k}^{−}|^{p})^{1/p} and Ω_{k}^{−} = {x ∈ ℝ^{N} : u_{k}(x) < 0}, we deduce from the equation satisfied by u_{k} that

\frac{1}{c_{k}} = ∫_{ℝ^{N}} n_{k}(v_{k})^{p} ≤ ∫_{Ω_{k}^{−}} n_{1k}(v_{k})^{p}.

Consider the case N > p (a similar argument holds in the case N ≤ p). We will argue as in the proof of Proposition 3.1. By Hölder inequality one has

\frac{1}{c_{k}} ≤ ∥n_{1k}∥_{L^{s}(Ω_{k}^{−})}∥v_{k}∥_{L^{p^{∗}} (ℝ^{N})}^{p} ≤ A^{p}∥n_{1k}∥_{L^{s}(Ω_{k}^{−})},

which implies that for some ε > 0, ∥n_{1k}∥_{L^{s}(Ω_{k}^{−})} ≥ ε for k sufficiently large. Moreover, since n_{1k} → n_{1} in L^{s}(ℝ^{N}), one can choose r > 0 so that ∥n_{1k}∥_{L^{s}(B_{r}^{c})}^{s} ≤ ε^{s}/2 for k sufficiently large, and consequently ∥n_{1k}∥_{L^{s}(Ω_{kr}^{−})}^{s} ≥ ε^{s}/2 where Ω_{kr}^{−} := Ω_{k}^{−}∩B_{r}.

Since n_{1k} converges to n_{1} in L_{loc}^{∞}(ℝ^{N}) one deduces from the latter relation that for some ζ > 0, |Ω_{kr}^{−}| > ζ for k sufficiently large. But this is impossible since u_{k} → φ_{m} in L^{p}(B_{r}) and φ_{m}(x) > 0 a.e. in B_{r}. □

Arguing as in Remark 2.3, one can deduce from Proposition 4.7 the following result.

Corollary 4.8. 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} ≥ 0, k = 1, 2, . . .. Assume that m_{1k}, n_{1k} belong to L_{loc}^{∞}(ℝ^{N}) ∩ L^{s}(ℝ^{N}) and converge in L^{s}(ℝ^{N}) to m_{1}, n_{1} respectively. Assume that m_{2k}, n_{2k} belong to L_{loc}^{∞}(ℝ^{N}) and converge in L_{loc}^{∞}(ℝ^{N}) to m_{2}, n_{2} respectively, with in addition, for some ε_{k} → 0,

|m_{2k} − m_{2}| ≤ ε_{k}m_{2} and |n_{2k} − n_{2}| ≤ ε_{k}n_{2} for a.e. x with |x| ≥ R. (4.9)

Then c(m_{k}, n_{k}) → c(m, n).

Remark 4.9. If (H_{4}) is replaced in Proposition 4.7 by (H′_{4}) (cf. Remark 2.5), then (4.5) can be replaced by the natural requirement that m_{2k}, n_{2k} converge in L^{N/p}(ℝ^{N}) to m_{2}, n_{2} respectively.

5. Fučik spectrum in 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 Fučik spectrum Σ was defined in the introduction (cf. (1.3)) and we will mainly consider here its part which lies in ℝ^{+} × ℝ^{+}. This part clearly contains the half lines λ_{1}(m) × ℝ^{+} and ℝ^{+} × λ_{1}(n). These half lines are in fact exactly made of those (α, β) ∈ Σ ∩ (ℝ^{+} × ℝ^{+}) for which (1.3) admits a solution which does not change sign. We denote below by Σ^{∗} the set Σ∩(ℝ^{+} ×ℝ^{+}) without these 2 trivial half lines. From the properties of the first eigenvalue recalled in Section 2, it easily follows that if (α, β) ∈ Σ^{∗}, then α > λ_{1}(m) and β > λ_{1}(n).

Theorem 5.1. For any r > 0, the line β = rα in the (α, β) plane intersects Σ^{∗}. Moreover the first point in this intersection is given by α(r) = c(m, rn), β(r) = rα(r), where c(., .) is defined in (3.7).

The proof of the above theorem is an easy consequence of Theorem 3.2 and Theorem 4.1

Letting r > 0 vary, we thus get a first curve C := {(α(r), β(r)) : r > 0} in Σ^{∗}. Here are some properties of this curve.

Proposition 5.2. The functions α(r) and β(r) in Theorem 5.1 are continuous. Moreover α(r) is strictly decreasing and β(r) is strictly increasing. One also has that α(r) → +∞ if r → 0 and β(r) → +∞ if r → +∞.

Proof. The continuity of the functions α(r) and β(r) follows directly from Proposition 4.7, and their strict monotonicity from Proposition 4.6. To show that α(r) → +∞ as r → 0, let us assume by contradiction that α(r) remains bounded as r → 0. Then β(r) = rα(r) → 0 as r → 0, which is impossible since β(r) > λ_{1}(n) for all r > 0. Similar argument for the behaviour of β(r) as r → +∞. □

We now investigate the asymptotic values α_{∞} := lim_{r→+∞} α(r) and β_{∞} := lim_{r→0} β(r) of the first curve in ℝ^{+} × ℝ^{+}. We will limit ourselves below to the study of α_{∞}; similar results can be proved for β_{∞} by interchanging m and n. The following lemma will be used. It is concerned with the eigenvalue problem

−Δ_{p}u = λm|u|^{p−2}u in B_{R}, u = 0 on ∂B_{R},  (5.1)

where B_{R} denotes the ball centred at the origin with radius R. Let λ_{R} denote the first positive eigenvalue of (5.1) and φ_{R} the associated positive eigenfunction such that ∫_{B_{R}} m(φ_{R})^{p} = 1 (which by (H_{3}) clearly exist for R sufficiently large ).

Lemma 5.3. λ_{R} → λ_{1}(m) and φ_{R} → φ_{m} in W as R → +∞.

Proof. We will adapt some arguments from [6, Lemma 5.2]. The function φ_{R} (extended by 0 outside B_{R}) clearly belongs to W and satisfies ∫_{ℝ^{N}} m(φ_{R})^{p} = 1. This implies

λ_{1}(m) ≤ ∫_{ℝ^{N}} |∇φ_{R}|^{p} = λ_{R},

and so lim inf λ_{R} ≥ λ_{1}(m). Let now δ ∈]0, 1[. Since φ_{m} ∈ W, there exists ψ ∈ C_{c}^{∞}(ℝ^{N}) such that

| ∫_{ℝ^{N}} (|∇φ_{m}|^{p} − |∇ψ|^{p})| ≤ \frac{δ}{2}, | ∫_{ℝ^{N}} m_{2}(φ_{m}^{p} − |ψ|^{p})| ≤ \frac{δ}{2},

| ∫_{ℝ^{N}} m_{1}(φ_{m}^{p} − |ψ|^{p})| ≤ \frac{δ}{2},

where we have used the imbedding of W into L^{p}(m_{1}, ℝ^{N}). This implies that | ∫_{ℝ^{N}} m(φ_{m}^{p}−|ψ|^{p})| ≤ δ and since ∫_{ℝ^{N}} m|φ_{m}|^{p} = 1, one deduces that ∫_{ℝ^{N}} m|ψ|^{p} > 0. We thus have, for R sufficiently large,

λ_{R} ≤ \frac{∫_{ℝ^{N}} |∇ψ|^{p}}{∫_{ℝ^{N}} m|ψ|^{p}} ≤ \frac{δ/2 + ∫_{ℝ^{N}} |∇φ_{m}|^{p}}{−δ + ∫_{ℝ^{N}} m|φ_{m}|^{p}} = \frac{δ/2 + λ_{1}(m)}{−δ + 1},

and since δ > 0 is arbitrary, we conclude that lim sup λ_{R} ≤ λ_{1}(m).

Let us now prove that φ_{R} → φ_{m} in W as R → +∞. One has

∫_{ℝ^{N}} |∇φ_{R}|^{p} = λ_{R} → λ_{1}(m) = ∫_{ℝ^{N}} |∇φ_{m}|^{p},  (5.2)

and so ∫_{ℝ^{N}} |∇φ_{R}|^{p} remains bounded. Moreover using the imbeddings of Section 2 as well as Lemma 2.2, one deduces from ∫_{ℝ^{N}} m(φ_{R})^{p} = 1 that ∫_{ℝ^{N}} m_{2}(φ_{R})^{p} remains bounded. Consequently φ_{R} remains bounded in W, and for a subsequence, φ_{R} → v weakly in W and strongly in L^{p}(m_{1}, ℝ^{N}). One has

∫_{ℝ^{N}} |∇v|^{p} ≤ lim inf ∫_{ℝ^{N}} |∇φ_{R}|^{p} = λ_{1}(m),

and also ∫_{ℝ^{N}} m|v|^{p} ≥ 1 (where the latter inequality follows from

∫_{ℝ^{N}} m_{2}|v|^{p} ≤ lim inf ∫_{ℝ^{N}} m_{2}(φ_{R})^{p} = −1 + ∫_{ℝ^{N}} m_{1}|v|^{p}).

The simplicity of λ_{1}(m) then implies v = φ_{m}. One also has

lim ∫_{ℝ^{N}} m_{2}(φ_{R})^{p} = lim [ − 1 + ∫_{ℝ^{N}} m_{1}(φ_{R})^{p}] = −1 + ∫_{ℝ^{N}} m_{1}φ_{m}^{p} = ∫_{ℝ^{N}} m_{2}φ_{m}^{p}.

(5.3)

Combining (5.2) and (5.3) with the weak convergence yields that φ_{R} → φ_{m} in W. □

The following proposition describes the asymptotic behaviour of the first curve C. Let us recall that the support of a measurable function u in ℝ^{N} is defined as the complement in ℝ^{N} of the largest open set on which u = 0 a.e.

Proposition 5.4. If N ≥ p, then α_{∞} = λ_{1}(m). If N < p, then α_{∞} = λ_{1}(m) if supp n^{+} is unbounded and α_{∞} > λ_{1}(m) if supp n^{+} is bounded.

Proof. One starts by introducing

\overline{α} := inf \{∫_{ℝ^{N}} |∇u^{+}|^{p} : u ∈ W, ∫_{ℝ^{N}} m(u^{+})^{p} = 1 and ∫_{ℝ^{N}} n(u^{−})^{p} > 0\} (5.4)

and showing that α_{∞} = \overline{α}. The proof of this equality is a direct adaptation of [2]. One also clearly has \overline{α} ≥ λ_{1}(m).

We first consider the case N ≥ p. In this case, the arguments of [2], which are local (they essentially involve approximating φ_{m} by a function which vanishes on a small ball where n^{+} ≢ 0), can be easily adapted to the present situation and give \overline{α} = λ_{1}(m).

We now consider the case where N < p and the support of n^{+} is unbounded. We will use the function φ_{R} defined in Lemma 5.3. Since supp n^{+} is unbounded, for any R > 0, n^{+} ≢ 0 on ℝ^{N}\\overline{B_{R}}. This allows by a regularization procedure to construct w_{R} ∈ C_{c}^{∞}(ℝ^{N}\\overline{B_{R}}) with w_{R} ≥ 0 and ∫_{ℝ^{N}} nw_{R}^{p} > 0. It follows that the function

u_{R} = φ_{R} − \frac{w_{R}}{R∥w_{R}∥_{W}}

converges to φ_{m} in W as R → +∞ and is admissible in the definition (5.4) of \overline{α}. Since by Lemma 5.3, ∫_{ℝ^{N}} |∇φ_{R}|^{p} → λ_{1}(m), we conclude that \overline{α} ≤ λ_{1}(m), and so \overline{α} = λ_{1}(m).

We finally consider the case where N < p and the support of n^{+} is bounded. Assume by contradiction \overline{α} = λ_{1}(m) and let u_{k} be a minimizing sequence for \overline{α}. Then ∫_{ℝ^{N}} |∇u_{k}^{+}|^{p} remains bounded and since ∫_{ℝ^{N}} m(u_{k}^{+}) = 1, arguing e.g. as in the proof of Lemma 5.3, one deduces that u_{k}^{+} remains bounded in W. Hence, for a subsequence, u_{k}^{+} → v weakly in W and strongly in L^{p}(m_{1}, ℝ^{N}). We then argue as in the second part of the proof of Lemma 5.3 to derive v = φ_{m}. Since φ_{m} ≥ some ε > 0 on the compact set supp n^{+}, it follows from the fact that u_{k}^{+} → φ_{m} uniformly on supp n^{+}, that u_{k}^{+} ≥ ε/2 on supp n^{+} for k sufficiently large. Consequently, for those k, u_{k}^{−} = 0 on supp n^{+}, which implies

∫_{ℝ^{N}} n(u_{k}^{−})^{p} = ∫_{ℝ^{N}} n^{+}(u_{k}^{−})^{p} − ∫_{ℝ^{N}} n^{−}(u_{k}^{−})^{p} = − ∫_{ℝ^{N}} n^{−}(u_{k}^{−})^{p} ≤ 0.

But this contradicts the fact that u_{k} is admissible in the definition (5.4) of \overline{α}. □

Remark 5.5. The distribution of Σ in the other quadrants of ℝ×ℝ can be studied in a manner similar to that used in [2]. It follows in particular that if N > p, m, n ∈ L^{N/p}(ℝ^{N}) ∩ L_{loc}^{∞}(ℝ^{N}), and m and n change sign, then Σ contains a first hyperbolic-like curve in each quadrant. The case N ≤ p however remains unclear (see in this respect Remark 2.1).

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Jean-Pierre Gossez Département de Mathématique, C. P. 214, Université Libre de Bruxelles, 1050 Bruxelles, Belgium

E-mail address: gossez@ulb.ac.be

Liamidi Leadi Institut de Mathématiques et de Sciences Physiques, Université d’Abomey Calavi, 01 BP : 613 Porto-Novo, Bénin Republic (West Africa)

E-mail address: leadiare@imsp-uac.org, leadiare@yahoo.com