2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 163–172. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE OF TWO NONTRIVIAL SOLUTIONS FOR SEMILINEAR ELLIPTIC PROBLEMS

ABDEL R. EL AMROUSS, FOUZIA MORADI, MIMOUN MOUSSAOUI

Abstract. This paper concerns the existence of multiple nontrivial solutions for some nonlinear problems. The first nontrivial solution is found using a minimax method, and the second by computing the Leray-Schauder index and the critical group near 0.

1. Introduction

We consider the Dirichlet problem

\begin{matrix}−Δu = λ_{k}u + f(u) in Ω \\ u = 0 on ∂Ω, \end{matrix}  (1.1)

where Ω is a bounded domain in ℝ^{n}, and f : Ω × ℝ → ℝ is a nonlinear function satisfying the Carathéodory conditions, and 0 < λ_{1} < λ_{2} ≤ … λ_{k} ≤ … is the sequence of eigenvalues of the problem

−Δu = λu in Ω,
u = 0 on ∂Ω.

Let us denote by E(λ_{j}) the λ_{j}-eigenspace and by F (s) the primitive ∫_{0}^{s} f(t) dt . There are several works studying the problem

\begin{matrix}−Δu = λ_{k}u + f(x, u) + h in Ω, \\ u = 0 on ∂Ω . \end{matrix}  (1.2)

where h ∈ L^{2}(Ω); see for example [4, 5, 6, 8, 9]. We write

l_{±}(x) = lim inf_{s→±∞} \frac{f(x, s)}{s}, k_{±}(x) = lim sup_{s→±∞} \frac{f(x, s)}{s},

L_{±}(x) = lim inf_{s→±∞} \frac{2F (x, s)}{s^{2}}, K_{±}(x) = lim sup_{s→±∞} \frac{2F (x, s)}{s^{2}} .

In [6], the solvability of (1.2) for every h ∈ L^{2}(Ω), is ensured when

0 < υ_{k} ≤ l_{±}(x) ≤ k_{±}(x) ≤ υ_{k+1} < λ_{k+1} − λ_{k},

2000 Mathematics Subject Classification. 58E05, 35J65, 56J20.

Key words and phrases. Variational elliptic problem; resonance; critical group;

Morse theory; minimax method.

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

Published September 20, 2006.

where υ_{k} and υ_{k+1} are constants.

However, in the autonomous case f(x, s) = f(s), De Figuerido and Gossez [5] introduced a density condition that requires \frac{f(s)}{s} to be between 0 and α = λ_{k+1}−λ_{k} as s → ±∞, and showed the existence of solution for any h. Next in [4], Costa and Oliveira proved an existence result for (1.2) under the following conditions:

0 ≤ l_{±}(x) ≤ k_{±}(x) ≤ λ_{k+1} − λ_{k} uniformly for a.e x ∈ Ω,  (1.3)
0 ⪯ L_{±}(x) ≤ K_{±}(x) ⪯ λ_{k+1} − λ_{k} uniformly for a.e x ∈ Ω.  (1.4)

Here the relation a(x) ⪯ b(x) indicates that a(x) ≤ b(x) on Ω, with strict inequality holding on subset of positive measure.

Later in [9], the authors proved an existence result in situation L_{±}(x) = 0 for a.e x ∈ Ω and K_{±}(x) = λ_{k+1} − λ_{k} for a.e x ∈ Ω. They replaced (1.4) by classical resonance conditions of Ahmad-Lazer-Paul on two sides of (1.4) and showed that (1.2) is solvable. More recently, in [8], the author interested to study the existence of two nontrivial solutions in the case k = 1 and under other weaker conditions cited above.

The aim of this paper is to generalize the above result for k ≥ 1. We assume the following assumptions:

(F0) |f′(s)| ≤ c(|s|^{p} + 1), s ∈ ℝ, p < \frac{4}{n−2} if n ≥ 3 and no restriction if n = 1, 2. (F1) sf(s) ≥ 0 for |s| ≥ r > 0 and

lim sup_{s→±∞} \frac{f(s)}{s} ≤ λ_{k+1} − λ_{k} = α.

(F2) lim_{∥v∥→∞,v∈E(λ_{k})} ∫ F (v(x))dx = +∞.

(F3) There exists η ∈ ℝ, 0 < η < α, such that

lim inf_{n→+∞} \frac{μ(G_{n})}{n} > 0

where G_{n} = \{s ∈] − n, n[, s ≠ 0, and \frac{f(s)}{s} ≤ α − η\} and μ denotes the Lebesgue measure on ℝ.

(F4) f′(0) + λ_{k} < λ_{1}

Theorem 1.1. Let f be C^{1} function, with f(0) = 0, that satisfies the conditions (F0)-(F4). Then (1.1) has at least two nontrivial solutions.

This paper is organized as follows: In section 2, we give some technical lemmas and some results of critical groups. The proof of our result is carried out in section 3.

2. Preliminaries Lemmas

Let us consider the functional defined on H_{0}^{1}(Ω) by

Φ(u) = \frac{1}{2} ∫_{Ω} |∇u|^{2}dx − \frac{1}{2}λ_{k} ∫ u^{2}dx − ∫ F (u)dx.

where H_{0}^{1}(Ω) is the usual Sobolev space obtained through the completion of C_{c}^{∞}(Ω) with respect to the norm induced by the inner product

⟨u, v⟩ = ∫_{Ω} ∇u∇vdx, u, v ∈ H_{0}^{1}(Ω).

It is well known that under a linear growth condition on f, the functional Φ is well defined on H_{0}^{1}(Ω), weakly lower semi-continuous and Φ ∈ C^{1}(H_{0}^{1}, ℝ), with

⟨Φ′(u), v⟩ = ∫_{Ω} ∇u∇vdx − λ_{k} ∫ uvdx − ∫ f(u)vdx, for u, v ∈ H_{0}^{1}(Ω).

Consequently, the weak solutions of the problem (1.1) are the critical points of the functional Φ. Moreover, under the condition(F0), Φ is a C^{2} functional with the second derivative given by

Φ′′(u)v.w = ∫ ∇v∇wdx − λ_{k} ∫ vwdx − ∫ f′(u)vwdx,

for u, v, w ∈ H_{0}^{1}(Ω).

Since we are going to apply the variational characterization of the eigenvalues, we shall decompose the space H_{0}^{1}(Ω) as E = E_{−} ⊕ E_{k} ⊕ E_{k+1} ⊕ E_{+}, where E_{−} is the subspace spanned by the λ_{j}- eigenfunctions with j < k and E_{j} is the eigenspace generated by the λ_{j}-eigenfunctions and E_{+} is the orthogonal complement of E_{−} ⊕ E_{k} ⊕ E_{k+1} in H_{0}^{1}(Ω) and we shall decompose for any u ∈ H_{0}^{1}(Ω) as following u = u^{−} + u^{k} + u^{+} where u^{−} ∈ E_{−}, u^{k} ∈ E_{k}, u^{k+1} ∈ E_{k+1} and u^{+} ∈ E_{+}. We can verify easily that

∫ |∇u|^{2} dx − λ_{i} ∫ |u|^{2} dx ≥ δ_{i}∥u∥^{2} ∀u ∈ ⊕_{j≥i+1}E_{j}  (2.1)
∫ |∇u|^{2} dx − λ_{i} ∫ |u|^{2} dx ≤ −δ_{i}∥u∥^{2} ∀u ∈ ⊕_{j≤i}E_{j},  (2.2)

where δ_{i} = min\{1 − \frac{λ_{i}}{λ_{i+1}} , \frac{λ_{i}}{λ_{i−1}} − 1\}.

2.1. A compactness condition. To apply minimax methods for finding critical points of Φ, we need to verify that Φ satisfies a compacteness condition of the Palais-Smail type which was introduced by Cerami [2], and recently was generalized by the first author in [7]. Definition. Let E be a real Banach space and Φ ∈ C^{1}(E, ℝ).

(i) A sequence (u_{n}) is said to be a (C)_{c} sequence, at the level c ∈ ℝ, if there is a sequence ϵ_{n} → 0, such that

Φ(u_{n}) → c  (2.3)
∥u_{n}∥⟨Φ′(u_{n}), v⟩_{H_{0}^{1},H^{−1}} ≤ ϵ_{n}∥v∥ ∀v ∈ H_{0}^{1}.  (2.4)

(ii) A functional Φ ∈ C^{1}(E, ℝ), is said to satisfy a condition (C)_{c}, at the level c ∈ ℝ, if every (C)_{c} sequence (u_{n}), possesses a convergent subsequence.

Now, we present some technical lemmas.

Lemma 2.1. Let (u_{n}) ⊂ H_{0}^{1}(Ω) and (p_{n}) ⊂ L^{∞}(Ω) be sequences, and let A a nonnegative constant such that

0 ≤ p_{n}(x) ≤ A a.e. in Ω and for all n ∈ ℕ

and p_{n} ⇀ 0 in the weak* topology of L^{∞}, as n → ∞. Then, there are subsequences (u_{n}), (p_{n}) satisfying the above conditions, and there is a positive integer n_{0} such that for all n ≥ n_{0},

∫ p_{n}u_{n}((u_{n}^{−} + u_{n}^{k}) − (u_{n}^{k+1} + u_{n}^{+})) dx ≥ \frac{−δ_{k}}{2} ∥u_{n}^{+} + u_{n}^{k+1}∥^{2}.  (2.5)

Proof. Since p_{n} ≥ 0 a.e. in Ω, we see that

\begin{matrix}∫ p_{n}u_{n}((u_{n}^{−} + u_{n}^{k}) − (u_{n}^{k+1} + u_{n}^{+})) \\ ≥ − ∫ p_{n}(u_{n}^{+} + u_{n}^{k+1})^{2} dx \\ ≥ −[ ∫ p_{n}(\frac{u_{n}^{+} + u_{n}^{k+1}}{∥u_{n}^{+} + u_{n}^{k+1}∥})^{2} dx]∥u_{n}^{+} + u_{n}^{k+1}∥^{2}. \end{matrix}  (2.6)

Moreover, by the compact imbedding of H_{0}^{1}(Ω) into L^{2}(Ω) and p_{n} ⇀ 0 in the weak* topology of L^{∞}, when n → ∞, then there are subsequences (u_{n}), (p_{n}) such that

∫ p_{n}(\frac{u_{n}^{+} + u_{n}^{k+1}}{∥u_{n}^{+} + u_{n}^{k+1}∥})^{2} dx → 0.

Therefore, there exists n_{0} ∈ ℕ such that for n ≥ n_{0} we have

∫ p_{n}(\frac{u_{n}^{+} + u_{n}^{k+1}}{∥u_{n}^{+} + u_{n}^{k+1}∥})^{2} dx ≤ \frac{δ_{k}}{2} .  (2.7)

Combining inequalities (2.6) and (2.7), we get inequality (2.5). □

Lemma 2.2. Let (u_{n}) ⊂ H_{0}^{1}(Ω) be a (C) sequence. If

f_{n}(x) = \frac{f(u_{n}(x))}{u_{n}(x)}χ_{[|u_{n}(x)|≥r_{ϵ}]} ⇀ 0

in the weak* topology of L^{∞}, as n → ∞. Then, there is subsequence (u_{n}) such that (∥u_{n}^{−} + (u_{n}^{+} + u_{n}^{k+1})∥)_{n} is uniformly bounded in n.

Proof. Since (u_{n})_{n} ⊂ H_{0}^{1} be a (C) sequence, (2.3) and (2.4) are satisfied. Now, we prove that the sequence (∥u_{n}^{−} + u_{n}^{+} + u_{n}^{k+1}∥)_{n} is uniformly bounded in n. Take v = (u_{n}^{−} + u_{n}^{k}) − (u_{n}^{+} + u_{n}^{k+1}) in (2.4), p_{n}(x) = f_{n}(x), and

Λ = { − ∫ |∇u_{n}^{−}|^{2} + λ_{k} ∫ |u_{n}^{−}|^{2} dx + ∫ |∇(u_{n}^{+} + u_{n}^{k+1})|^{2}
− λ_{k} ∫ |u_{n}^{+} + u_{n}^{k+1}|^{2} dx + ∫ p_{n}u_{n}((u_{n}^{−} + u_{n}^{k}) − (u_{n}^{k+1} + u_{n}^{+})) dx}
Γ = {ϵ_{n} + ∫_{|u_{n}(x)|≤r_{ϵ}} |f(u_{n}(x)||(u_{n}^{+} + u_{n}^{k+1}) − (u_{n}^{−} + u_{n}^{k})| dx}.

Then Λ ≤ Γ. By the Poincaré inequality, from (2.1 ), (2.2), (2.5), and Λ ≤ Γ, it follows that there exists constants A_{ϵ} and B_{ϵ} such that

\frac{δ_{k}}{2} ∥u_{n}^{−} + (u_{n}^{+} + u_{n}^{k+1})∥^{2} ≤ ϵ_{n} + A_{ϵ}∥u_{n}^{−} + (u_{n}^{+} + u_{n}^{k+1})∥ + B_{ϵ}.

This gives that (∥u_{n}^{−} + (u_{n}^{+} + u_{n}^{k+1})∥)_{n} is uniformly bounded in n. □

Lemma 2.3. Let (u_{n}) ⊂ H_{0}^{1}(Ω) such that ∥u_{n}^{−} +(u_{n}^{+} +u_{n}^{k+1})∥ is uniformly bounded in n and there exists A such that if A ≤ Φ(u_{n}), then

∫ F (\frac{u_{n}^{k}}{2} ) dx ≤ M.

Proof. From A ≤ Φ(u_{n}), and Poincaré inequality, we have

∫ F (\frac{u_{n}^{k}}{2} )dx ≤ −A + ∫ [F (\frac{u_{n}^{k}}{2} ) − F (u_{n})]dx + \frac{1}{2}∥u_{n}^{−} + u_{n}^{+} + u_{n}^{k+1}∥^{2}.  (2.8)

Since f ∈ C^{1} (\overline{Ω}, ℝ) satisfy (F1), there exists two functions γ, h : Ω → ℝ such that

f(t) = tγ (t) + h (t)

with 0 ≤ γ (t) = \frac{f(t)}{t}χ[|t| ≥ r] ≤ λ_{k+1} − λ_{k} and h(t) = f(t)χ[|t| < r]. However, by the mean value theorem, we get

∫ [F (\frac{u_{n}^{k}}{2} ) − F (u_{n})]dx = ∫_{Ω} ∫_{0}^{1} f(t\frac{u_{n}^{k}}{2} + (1 − t)u_{n})dt(\frac{u_{n}^{k}}{2} − u_{n})dx

= ∫_{Ω} ∫_{0}^{1} h(t\frac{u_{n}^{k}}{2} + (1 − t)u_{n})dt(\frac{u_{n}^{k}}{2} − u_{n})dx

+ ∫_{Ω} ∫_{0}^{1} γ(t\frac{u_{n}^{k}}{2} + (1 − t)u_{n})[t(\frac{u_{n}^{k}}{2} − u_{n})^{2} + (\frac{u_{n}^{k}}{2} − u_{n})u_{n}]

(2.9)

Set t_{1} = min\{t ∈ [0, 1] : ∫_{0}^{1} h(t\frac{u_{n}^{k}}{2} + (1 − t)u_{n}) ≠ 0\} and t_{2} = max\{t ∈ [0, 1] :

∫_{0}^{1} h(t\frac{u_{n}^{k}}{2} + (1 − t)u_{n}) ≠ 0\}. It is clear that

(t_{2} − t_{1})|\frac{u_{n}^{k}}{2} − u_{n}| ≤ 2r.  (2.10)

So that using (2.9),(2.10) and the Poincaré inequality, and an elementary inequality

(\frac{a}{2} − b)^{2} + (\frac{a}{2} − b)b ≤ (a − b)^{2}.

We have

∫ [F (\frac{u_{n}^{k}}{2} ) − F (u_{n})]dx

≤ ∫_{Ω} ∫_{t_{1}}^{t_{2}} h(t\frac{u_{n}^{k}}{2} + (1 − t)u_{n})dt(\frac{u_{n}^{k}}{2} − u_{n})dx + \frac{λ_{k+1} − λ_{k}}{4λ_{1}} ∥u_{n}^{−} + u_{n}^{+} + u_{n}^{k+1}∥^{2}

≤ 2r sup_{|s|≤r} |f(s)| meas(Ω) + \frac{λ_{k+1} − λ_{k}}{4λ_{1}} ∥u_{n}^{−} + u_{n}^{+} + u_{n}^{k+1}∥^{2}.

(2.11)

From (2.8) and (2.11), there exists M > 0 such that

∫ F (\frac{u_{n}^{k}}{2} ) dx ≤ M.

□

2.2. Critical groups. Let H be a Hilbert space and Φ ∈ C^{1}(H, ℝ) satisfying the Palais-Smaile condition or the Cerami condition. Set Φ^{c} = {u ∈ H | Φ(u) ≤ c} and denote by H_{q}(X, Y) the q-th relative singular homology group with integer coefficient. The critical groups of Φ at an isolated critical point u with Φ(u) = c are defined by

C_{q}(Φ, u) = H_{q}(Φ^{c} ∩ U, Φ^{c} ∩ U \ {u}); q ∈ Z.

where U is a closed neighborhood of u.

Let K = {u ∈ H | Φ′(u) = 0} be the set of critical points of Φ and a < inf_{K} Φ. The critical groups of Φ at infinity are defined by

C_{q}(Φ, ∞) = H_{q}(H, Φ^{a}); q ∈ Z

We will use the notation deg(Φ′, U, 0) for the Leray-Schauder degree of Φ with respect to the set U and the value 0. Denote also by index(Φ′, u) the Leray-Schauder index of Φ′ at critical point u. Recall that this quantity is defined as lim_{r→0} deg(Φ′, B_{r}(u), 0), if this limit exists, where B_{r}(u) is the ball of radius r centered at u.

Proposition 2.4 ([3]). If u is a mountain pass point of Φ, then

C_{q}(Φ, u) = δ_{q,1}Z.

Proposition 2.5 ([1]). Assume that H = H^{+} ⊕ H^{−}, Φ is bounded from below on H^{+} and Φ(u) → −∞ as ∥u∥ → ∞ with u ∈ H^{−}. Then

C_{μ}(Φ, ∞) ≠ 0, with μ = dim H^{−} < ∞.

3. Proof of Theorem 1.1

First, we prove that Φ satisfies the Cerami condition.

Lemma 3.1. Under the assumptions (F0)–(F3), Φ satisfies the (C)_{c} condition on H_{0}^{1}(Ω), for all c ∈ ℝ.

Proof. Let (u_{n})_{n} ⊂ H_{0}^{1} be a (C)_{c} sequence, i.e

Φ(u_{n}) → c  (3.1)
∥u_{n}∥⟨Φ′(u_{n}), v⟩_{H_{0}^{1},H^{−1}} ≤ ϵ_{n}∥v∥ ∀v ∈ H_{0}^{1},  (3.2)

where ϵ_{n} → 0. It clearly suffices to show that (u_{n})_{n} remains bounded in H_{0}^{1}. Assume by contradiction. Defining z_{n} = \frac{u_{n}}{∥u_{n}∥}, we have ∥z_{n}∥ = 1 and, passing if necessary to a subsequence, we may assume that z_{n} ⇀ z weakly in H_{0}^{1}, z_{n} → z strongly in L^{2}(Ω) and z_{n}(x) → z(x) a.e. in Ω. By the linear growth of f , the sequence (\frac{f(u_{n}(x))}{∥u_{n}∥})_{n} remains bounded in L^{2}, then for a subsequence, we have

\frac{f(u_{n}(x))}{∥u_{n}∥} ⇀ ζ in L^{2}.

and by standard arguments based on assumptions F0),F1), ζ can be written as ζ(x) = m(x)z(x), where m satisfies (see [4]).

0 ≤ m(x) ≤ λ_{k+1} − λ_{k} a.e. in Ω.

However, divide (3.2) by ∥u_{n}∥^{2} and goes to the limit we obtain

\frac{⟨Φ′(u_{n}), v⟩}{∥u_{n}∥} = ∫ ∇z_{n}∇v − λ_{k} ∫ z_{n}v − ∫ \frac{f(u_{n})}{∥u_{n}∥}vdx → 0

for every v ∈ H_{0}^{1}. On the other hand, since z_{n} converges to z weakly in H_{0}^{1}, strongly in L^{2} and \frac{f(u_{n}(x))}{∥u_{n}∥} converges weakly in L^{2} to ζ, we deduce

\frac{⟨Φ′(u_{n}), v⟩}{∥u_{n}∥} → ∫ ∇z∇v − λ_{k} ∫ zv − ∫ ζvdx = 0 ∀v ∈ H_{0}^{1}(Ω).  (3.3)

Claim: We will prove that z_{n} → z strongly in H_{0}^{1}. Indeed, taking v = z in (3.3) we have

∥z∥^{2} = λ_{k} ∫ z^{2} + ∫ m(x)z^{2}.  (3.4)

On the other hand, by (3.2) it results

\frac{⟨Φ′(u_{n}), u_{n}⟩}{∥u_{n}∥^{2}} → 1 − λ_{k} ∫ z^{2} − ∫ m(x)z^{2} = 0.  (3.5)

From (3.4) and (3.5), it follows ∥z∥ = 1. Since z_{n} ⇀ z, ∥z_{n}∥ → ∥z∥ and H_{0}^{1}(Ω) is convex uniformly space the claim follows. So that, z is a nontrivial solution of problem

\begin{matrix}−Δz = (λ_{k} + m(x))z in Ω \\ z = 0 on ∂Ω. \end{matrix}  (3.6)

We now distinguish three cases: i) λ_{k} < m(x)+λ_{k} and m(x)+λ_{k} < λ_{k+1} on subset of positive measure; (ii) m(x) + λ_{k} ≡ λ_{k}; (iii) m(x) + λ_{k} ≡ λ_{k+1}.

Case i: We have z is a nontrivial solution of problem (3.6), then 1 is an eigenvalue of this problem. On the other hand, by strict monotonicity λ_{k} (λ_{k} + m(x)) < 1 and λ_{k+1} (λ_{k} + m(x)) > 1 , which gives a contradiction.

Case ii: By (F1), for ε > 0, there exists a constant r_{ε} > r such that

0 ≤ \frac{f(s)}{s} ≤ λ_{k+1} − λ_{k} + ε ∀|s| ≥ r_{ε}  (3.7)

Put f_{n}(x) = ^{\frac{f(u_{n}(x))}{u_{n}(x)}}χ\{|u_{n}(x)| ≥ r_{ε}\}, which remains bounded in L^{∞}, passing if necessary to a subsequence, f_{n} → l in the weak* topology of L^{∞}. By (3.7), the L^{∞}-function l satisfies

0 ≤ l(x) ≤ λ_{k+1} − λ_{k} + ε a.e.in Ω

Multiply f_{n} by z_{n}^{2}, integrate on Ω and going to the limit, to have

∫ f_{n}z_{n}^{2}dx = ∫ \frac{f (u_{n}(x))}{∥u_{n}∥}z_{n} → ∫ m(x)z^{2}dx = ∫ l(x)z^{2}dx = 0.

By the unique continuation Property of Δ and l ≥ 0, we deduce that l ≡ 0 a.e.in Ω. Then, by lemma 2.2 and lemma 2.3 there exists M > 0 such that

∫ F (\frac{u_{n}^{k}}{2} )dx ≤ M.

This is a contradiction with assumption (F2) and ∥u_{n}^{k}∥ → +∞.

Case iii: If m(x) ≡ λ_{k+1} − λ_{k}. Dividing (3.1) by ∥u_{n}∥^{2}, we obtain

\frac{Φ(u_{n})}{∥u_{n}∥^{2}} = \frac{1}{2}∥z_{n}∥^{2} − \frac{λ_{k}}{2} ∫ z_{n}^{2} − ∫ \frac{F (u_{n}(x))}{∥u_{n}∥^{2}}dx → 0, as n → ∞.

However, it results that

lim_{n→+∞} ∫ \frac{F (u_{n}(x))}{∥u_{n}∥^{2}}dx = \frac{1}{2}α ∫ z^{2}dx.

Applying Fatou’s lemma, we have

∫_{z>0}(α − K_{+})z^{2}dx + ∫_{z<0} (α − K_{−}) z^{2}dx ≤ 0.

This is a contradiction with assumption (F3), since (F3) is equivalent to K_{±} = lim sup_{s→±∞} \frac{2F (s)}{s^{2}} < α. (see [10]). The proof of lemma is complete. □

Lemma 3.2. Under the hypothesis of Theorem 1.1, the functional Φ has the following properties:

(i) Φ(w) → +∞, as ∥w∥ → +∞, w ∈ W^{+} = E_{k+1} ⊕ E_{+}.

(ii) Φ(v) → −∞, as ∥v∥ → +∞, w ∈ W^{−} = E_{k} ⊕ E_{−}.

Proof. (i) Φ is coercive on W^{+}. Indeed, the assumption (F3) is equivalent to K_{±} = lim sup_{s→±∞} \frac{2F (s)}{s^{2}} < α. Thus, there exists an B_{ε} ≥ 0 such that

F (s) ≤ \frac{α}{2}s^{2} − εs^{2} + B_{ε} ∀s ∈ ℝ.

Hence, for every w ∈ W^{+}, we obtain

Φ(w) = \frac{1}{2}∥w∥^{2} − \frac{λ_{k}}{2} ∫ w^{2} − ∫ F (w) dx

≥ \frac{λ_{k+1} − λ_{k}}{2λ_{k+1}} ∥w∥^{2} − \frac{α − 2ε}{2} ∫ w^{2} − B_{ε}|Ω|

≥ \frac{ε}{λ_{k+1}} ∥w∥^{2} − B_{ε}|Ω|.

However, Φ(w) → +∞, as ∥w∥ → +∞. (ii) Assume by contradiction that there exists a constant B > 0 and a sequence (v_{n}) ⊂ V with ∥v_{n}∥ → ∞ such that

B ≤ Φ(v_{n}) ≤ −δ∥v_{n}^{−}∥^{2}.

Therefore, by lemma 2.3, since ∥v_{n}^{−}∥ is bounded, there exists M > 0 such that

∫ F (\frac{v_{n}^{k}}{2} ) dx ≤ M

which contradicts (F2). □

Lemma 3.3. Under the condition (F4), the functional Φ has the following properties:

(i) There is an R > 0 and β > 0 such that Φ ≥ β on ∂B_{R}(0).

(ii) C_{q}(Φ, 0) = δ_{q,0}Z

Proof. (i) We start by proving the first assertion. On one hand, it is easy to see that if λ_{k} + f′(0) ≤ 0 we have

Φ′′(0)u.u ≥ ∥u∥^{2}.

On the other hand, where λ_{k} + f′(0) > 0, the Poincaré’s inequality gives that

Φ′′(0)u.u = ∥u∥^{2} − λ_{k} ∫ u^{2} − ∫ f′(0)u^{2}dx ≥ (1 − \frac{λ_{k} + f′(0)}{λ_{1}})∥u∥^{2}

Put γ = 1 − \frac{λ_{k}+f′(0)}{λ_{1}} and by (F4), we have γ > 0 and

Φ′′(0)u.u ≥ γ∥u∥^{2}.

Taylor’s formula implies

Φ(u) = \frac{1}{2}Φ′′(0)u.u + o(∥u∥^{2}) ≥ (\frac{γ}{2} + \frac{o(∥u∥^{2})}{∥u∥^{2}})∥u∥^{2}

with \frac{o(∥u∥^{2})}{∥u∥^{2}} → 0, as ∥u∥ → 0. Consequently, the assertion (i) follows. (ii) Since u = 0 is a local mininum of Φ, we have

C_{q}(Φ, 0) = δ_{q,0}Z.
□

Lemma 3.4. The functional Φ has at least one critical point u_{0}, such that

C_{q}(Φ, u_{0}) = δ_{q,1}Z.

Proof. According to (ii) of Lemma 3.2, Φ is anti-coercive on W^{−} we can find an e ∈ H_{0}^{1} such that ∥e∥ ≥ M > R andΦ(e) ≤ 0. So by mountain pass theorem, there exists a critical point u_{0} of mountain pass type, such that

C_{1}(Φ, u_{0}) ≠ 0.

By proposition 2.4, it results that C_{q}(Φ, u_{0}) = δ_{q,1}Z. The proof of lemma is complete. □

Proof of Theorem 1.1. For this proof we distinguish two cases.

Case 1: If k = 1, we assume that {0, u_{0}} is the critical set of Φ and let R > 0, such that {0, u_{0}} ⊂ B_{R}(0). By the Riesz representation theorem we can write

⟨Φ′(u), v⟩ = ⟨u, v⟩ − ⟨Nu, v⟩, for all u, v ∈ H_{0}^{1}(Ω)

where ⟨u, v⟩ = ∫_{Ω} ∇u∇v and ⟨Nu, v⟩ = ∫ [λ_{1}u+f(u)]v dx. So that, Φ′ = I −N and By the Sobolev embedding theorem, N is compact. We see that Φ′ has the form Identity-compact, so that Leary-Shauder techniques are applicable

deg(Φ′, B_{R}(0), 0) = index(Φ′, 0) + index(Φ′, u_{0})

= ∑_{q=0}^{∞}(−1)^{q} dim C_{q} (Φ, 0) + ∑_{q=0}^{∞}(−1)^{q} dim C_{q} (Φ, u_{0})

= 1 − 1 = 0

(3.8)

In a similar way we can define a compact map T : H_{0}^{1}(Ω) → H_{0}^{1}(Ω) by

⟨T u, v⟩ = ∫ (λ_{1} + μ)uv dx

where 0 < μ < λ_{2} − λ_{1}. Now we claim that there is a priori bound in H_{0}^{1}(Ω) for all possible solutions of the family of equations (see [12])

−Δu − λ_{1}u = (1 − t)μu + tf(u) in Ω
u = 0 on ∂Ω.

The homotopy invariance of Leray-Schauder degree implies

deg(Φ′, B_{R}(0), 0) = deg(I − T, B_{R}(0), 0) = −1.

This contradicts (3.8).

Case 2: If k ≥ 2, by Lemma 3.1, the functional Φ satisfies the condition (C). Since Φ is weakly lower semi continuous and coercive on W^{+}, Φ is bounded from below on W^{+}. Moreover, by (ii) of Lemma 3.2, Φ is anti-coercive on W^{−}, thus we can apply the proposition 2.5 and we conclude that

C_{μ}(Φ, ∞) ≠ 0

where μ = dim W^{−} ≥ k ≥ 2. It follows from the Morse inequality that Φ has a critical point u_{1} with

C_{μ}(Φ, u_{1}) ≠ 0.

Since μ ≠ 1 and μ ≠ 0, then the problem (1.1) has at least two nontrivial solutions. The proof of theorem is complete. □

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Abdel R. El Amrouss University Mohamed 1er, Faculty of sciences, Department of Mathematics, Oujda, Morocco

E-mail address: amrouss@sciences.univ-oujda.ac.ma

Fouzia Moradi University Mohamed 1er, Faculty of sciences, Department of Mathematics, Oujda, Morocco

E-mail address: foumoradi@yahoo.fr

Mimoun Moussaoui University Mohamed 1er, Faculty of sciences, Department of Mathematics, Oujda, Morocco

E-mail address: moussaoui@sciences.univ-oujda.ac.ma