2002-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 09, 2002, pp 93–106. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Existence and multiplicity of nontrivial solutions for double resonance semilinear elliptic problems ^{∗}

Abdel R. El Amrouss

Abstract

We consider resonance problems at an arbitrary eigenvalue of the Laplacien. We prove the existence of nontrivial solutions for some semilinear elliptic Dirichlet boundary values problems. We also obtain two nontrivial solutions by using Morse theory.

1 Introduction

Let Ω be a bounded domain in ℝ^{n}, and let g : Ω×ℝ → ℝ be a nonlinear function satisfying the Carathéodory conditions. We consider the Dirichlet problem

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

where 0 < λ_{1} < λ_{2} ≤ … λ_{k} ≤ … is the sequence of eigenvalues of the problem

−Δu = λu in Ω,
u = 0 on ∂Ω .

Let F (x, s) be the primitive ∫_{0}^{s} f(x, t) dt, and

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

Let us assume that 0 ≤ l_{±}(x) ≤ k_{±}(x) ≤ λ_{k+1} − λ_{k} uniformly for a.e. x ∈ Ω.

There have been many papers concerning problem (1.1) at resonance in the situation where l_{±}(x) ≡ 0 or k_{±}(x) ≡ λ_{k+1} − λ_{k}; see for example [11, 1, 4, 9, 8, 17]. Some multiplicity theorems are obtained by using the topological degree technique and the variational methods, [2, 12, 13, 14, 5, 10, 7, 16].

In this paper, we are interested in finding nontrivial solutions of (1.1). First, we prove the existence of a nontrivial solution when f(x, s)/s stays between 0 and λ_{k+1} − λ_{k} for large values of |s|. We replace the non-resonance conditions of Costa-Oliviera [8] by classical resonance conditions of Ahmad-Lazer-Paul.

∗Mathematics Subject Classifications: 58E05, 35J65, 56J20.

Key words: Variational elliptic problem, resonance, critical group, Morse theory,

minimax method.

\begin{matrix}c \\ ◯ \end{matrix}2002 Southwest Texas State University.

Published December 28, 2002.

Let us denote by E(λ_{j}) the λ_{j}-eigenspace, and state the following hypotheses:

(F0) For all R > 0, sup_{|s|≤R} |f(x, s)| ∈ L^{∞}(Ω).

(F1) 0 ≤ f(x, s)s ≤ (λ_{k+1} − λ_{k})s^{2} for |s| ≥ r > 0 and a.e. x ∈ Ω

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

(F3) lim_{∥u∥→+∞,u∈E(λ_{k+1})} ∫ [\frac{1}{2}(λ_{k+1} − λ_{k})u^{2}(x) − F (x, u(x))] dx = +∞.

(F4) lim sup_{s→±∞} \frac{2F (x, s)}{s^{2}} ≤ β < λ_{1} − λ_{k} uniformly for a.e. x ∈ Ω.

Now, we state the following result.

Theorem 1.1 Under the assumptions (F0)–(F4), Problem (1.1) has at least one nontrivial solution.

Note that (F4) implies f(x, 0) = 0 for a.e. x ∈ Ω, so that (1.1) has the trivial solution in this case.

The second purpose of this paper is to study the existence of at least two nontrivial solutions of (1.1) when f is C^{1}, f(x, s)s lies between 0 and (λ_{k+1} − λ_{k})s^{2} for large values of |s|, and

(F5) f(x, s)s → +∞, uniformly on Ω, as s → +∞ or as s → −∞.

(F6) (λ_{k+1} − λ_{k})s^{2} − sf(x, s) → +∞, uniformly on Ω, as s → +∞ or as s → −∞.

(F7) f′(x, 0) + λ_{k} ≤ λ_{1} on Ω, with strict inequality f′(x, 0) + λ_{k} < λ_{1} holding on subset of positive measure.

Our main results now read as the follows.

Theorem 1.2 Suppose that f ∈ C^{1}(\overline{Ω} × ℝ, ℝ) such that f(x, 0) = 0 for a.e. x ∈ Ω, and (F5)–(F7) are satisfied. Then (1.1) has at least two nontrivial solutions.

As an immediate consequence we obtain the corollary below, under the assumption that

(F8) 0 < f′(s) < λ_{k+1} − λ_{k} for |s| ≥ r > 0.

Corollary 1.3 Assume that f(x, s) = f(s) ∈ C^{1}(ℝ), f(0) = 0, f′(0) + λ_{k} < λ_{1} and (F8) is satisfied. Then, (1.1) has at least two nontrivial solutions.

Theorem 1.4 Suppose that f ∈ C^{1}(\overline{Ω} × ℝ, ℝ) such that f(x, 0) = 0 for a.e. x ∈ Ω. Assume that f satisfies (F5), (F6), k = 1, and (F8). Also assume that there is an m ≥ 2 such that

λ_{m} ≤ f′(x, 0) + λ_{1} ≤ λ_{m+1} on Ω,

with strict inequality λ_{m} < f′(x, 0) + λ_{1} < λ_{m+1} holding on subset of positive measure. Then, (1.1) has at least two nontrivial solutions.

Remarks

1. It is obvious to see that the conditions (F5)-(F6) imply the conditions (F2)-(F3).

2. Note that our multiplicity results are not covred by the results mentioned in [2, 12, 13, 14, 5, 10]. In fact, the condition sup_{t∈ℝ} f′(t) < λ_{k+1} − λ_{k} cited in [13, 5, 10] implies (F6).

The proof of Theorem 1.1 uses a variational argument and the general minimax theorem proved by Bartolo in [3]. In section 4, using Morse theory we compare the computed critical groups of Φ at the trivial critical point and the first nontrivial critical point given by minimax method. The existence of the second nontrivial solution is deduced from the calculation of the Leray-Schauder index of critical points.

2 Preliminaries

By a solution of (1.1) we mean a function u ∈ H_{0}^{1}(Ω) satisfying

∫_{Ω} ∇u∇v − λ_{k} ∫_{Ω} uv − ∫_{Ω} f(x, u)v = 0, forallv ∈ H_{0}^{1}(Ω)

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

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

For u ∈ H_{0}^{1}(Ω) define the functional

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

It is well know that under a sub-critical growth condition on f, Φ is well defined on H_{0}^{1}(Ω), weakly lower semi-continuous and continuously Fréchet differentiable, with derivative

Φ′(u)v = ∫_{Ω} ∇u∇v − λ_{k} ∫ uv − ∫ f(x, u)v, for u, v ∈ H_{0}^{1}(Ω)

Thus, finding solutions of (1.1) is equivalent to finding critical points of the functional Φ.

To apply minimax methods for finding critical points of Φ, we need to verify that Φ satisfies a compactness condition of the Palais-Smale type which was introduced by Cerami.

Definition A functional Φ ∈ C^{1}(E, ℝ), with E a real Banach space, is said to satisfy condition (C)_{c}, at the level c ∈ ℝ, if: Every sequence (u_{n}) ⊂ E such that

Φ(u_{n}) → c, ∥u_{n}∥Φ′(u_{n}) → 0

possesses a convergent subsequence

It was shown in [3] that condition (C)_{c} actually suffices to get a deformation theorem. Then, by standard minimax arguments [3], the following result was proved.

Theorem 2.1 Suppose that Φ ∈ C^{1}(E, ℝ), E a real Banach space, satisfies condition (C)_{c}∀c ∈ ℝ and that there exist a closed subset S ⊂ E and Q ⊂ E with boundary ∂Q satisfying the following conditions

i) sup_{u∈∂Q} Φ(u) ≤ α < β ≤ inf_{u∈S} Φ(u) for some 0 ≤ α < β

ii) The intersection of S and ∂Q is not empty and for every h ∈ C(E, E) such that h/∂Q = Id, we have h(Q) ∩ S ≠ ∅.

iii) sup_{u∈Q} Φ(u) < ∞.

Then Φ possesses a critical value c ≥ β.

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\}.

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}.  (2.3)

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}  (2.4)

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} .  (2.5)

Combining inequalities (2.4) and (2.5), we get inequality (2.3). □

Definition 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  (2.6)
∥u_{n}∥⟨Φ′(u_{n}), v⟩_{H_{0}^{1},H^{−1}} ≤ ϵ_{n}∥v∥ ∀v ∈ H_{0}^{1}.  (2.7)

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

1. If f_{n}(x) = \frac{f(x,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.

2. If f_{n}(x) = \frac{(λ_{k+1}−λ_{k})u_{n}(x)−f(x,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})∥)_{n} is uniformly bounded in n.

Proof: Since (u_{n})_{n} ⊂ H_{0}^{1} be a (C) sequence, (2.6) and (2.7) 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 (8), 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} + ∫ h((u_{n}^{+} + u_{n}^{k+1}) − u_{n}^{−}) dx
+ ∫_{|u_{n}(x)|≤r_{ϵ}} |f(x, 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), (2.3), and Λ ≤ Γ, it follows that there exist 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. The same proof of eventuality 2 are given by taking v = (u_{n}^{+} + u_{n}^{k+1}) − (u_{n}^{−} + u_{n}^{k}) and

p_{n}(x) = f_{n}(x) = \frac{(λ_{k+1} − λ_{k})u_{n}(x) − f(x, u_{n}(x))}{u_{n}(x)}χ_{[|u_{n}(x)|≥r_{ϵ}]}.

Lemma 2.3 1. 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 (x, \frac{u_{n}^{k}}{2} ) dx ≤ M.

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

∫ [\frac{λ_{k+1} − λ_{k}}{2} (\frac{u_{n}^{k+1}}{2} )^{2} − F (x, \frac{u_{n}^{k+1}}{2} )] dx ≤ M.

Proof: ¿From (2.6), and Poincaré inequality, we have

\begin{matrix}∫ F (x, \frac{u_{n}^{k}}{2} ) dx ≤ & A + ∫ [F (x, \frac{u_{n}^{k}}{2} ) − F (x, u_{n})] dx \\ + \frac{1}{2}∥u_{n}^{+} + u_{n}^{k+1} + u_{n}^{−}∥^{2} + \frac{1}{\sqrt{λ_{1}}} ∥h∥_{L^{2}}∥u_{n}^{+} + u_{n}^{k+1} + u_{n}^{−}∥. \end{matrix}

(2.8)

However, by the mean value theorem, we get for a.e. x ∈ Ω an t = t(x) ∈ [0, 1] such that

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

= ∫ f(x, t\frac{u_{n}^{k}}{2} + (1 − t)u_{n})(\frac{u_{n}^{k}}{2} − u_{n}) dx

= ∫_{|t \frac{u^{k}_{n}}{2} +(1−t)u_{n}|≤rϵ} f(x, t\frac{u_{n}^{k}}{2} + (1 − t)u_{n}) dx

+ ∫_{|t \frac{u^{k}_{n}}{2} +(1−t)u_{n}|≥rϵ} \frac{f(x, t\frac{u_{n}^{k}}{2} + (1 − t)u_{n})}{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}] dx.

(2.9)

So that using (2.9) and the Poincaré inequality again we have

\begin{matrix}∫ [F (x, \frac{u_{n}^{k}}{2} ) − F (x, u_{n})] dx \\ ≤ \frac{2}{\sqrt{λ_{1}}} ∥ sup_{|s|≤rϵ} |f(x, s)|∥_{L^{2}}∥u_{n}^{−} + u_{n}^{+} + u_{n}^{k+1}∥ \\ + r_{ϵ}∥ sup_{|s|≤rϵ} |f(x, s)|∥_{L^{1}} + \frac{λ_{k+1} − λ_{k} + ϵ}{4λ_{1}}∥u_{n}^{−} + u_{n}^{+} + u_{n}^{k+1}∥^{2}. \end{matrix}  (2.10)

¿From (2.8) and (2.10), there exists M > 0 such that

∫ F (x, \frac{u_{n}^{k}}{2} ) dx ≤ M.

3 Proof of Theorem 1.1

To apply Theorem 2.1, we shall do separate studies of the “compactness” of Φ and its “geometry”. First, we prove that Φ satisfies the Cerami condition.

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

where A is a constant and 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. As in the proof of Lemma 2.1, we put

f_{n}(x) = \frac{f(x, u_{n}(x))}{u_{n}(x)}χ_{[|u_{n}(x)|≥r]}

and l ∈ L^{∞} such that f_{n} → l in the weak* topology of L^{∞}, as n → ∞. where the L^{∞}-function l satisfies

0 ≤ l(x) ≤ λ_{k+1} − λ_{k}.  (3.1)

Set m(x) = l(x) + λ_{k}, by (2.6), we have

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

So that, z ≢ 0. In other words , we verify easily that z satisfies

\begin{matrix}−Δz = m(x)z in Ω \\ z = 0 on ∂Ω \end{matrix}  (3.2)

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

In case i), we have λ_{k}(m) < 1 and λ_{k+1}(m) > 1. This contradicts that 1 is an eigenvalue of problem (3.2). On the other hand, by (F2), (F3), lemmas 2.2 and 2.3 the cases ii) and iii) are not possible. The proof is complete. □

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

i) Φ(v) → −∞, as ∥v∥ → ∞, v ∈ E_{k} ⊕ E_{−} = V

ii There is an r > 0 such that Φ ≤ α on ∂B_{r}(0).

Proof i) Assume by contradiction there exist a constant B and a sequence (v_{n}) ⊂ V with ∥v_{n}∥ → ∞ such that

B ≤ Φ(v_{n}) ≤ −δ∥v_{n}^{−}∥^{2}.

Therefore, ∥v_{n}^{−}∥ is bounded and by Lemma 2.3, we obtain

lim inf_{n→∞} ∫ F (x, \frac{v_{n}^{k}}{2} ) dx ≤ constant.

This is a contradiction with assumption (F2). ii) By (F4), there exists δ > 0 such that

F (x, s) ≤ \frac{β′}{2} |s|^{2} for |s| ≤ δ and a.e. x ∈ Ω  (3.3)

where β′ ∈]β, λ_{1} − λ_{k}[.

On the other hand, (F0) and (F1) implies that there exist μ > 2 and A > 0 such that

F (x, s) ≤ A|s|^{μ} for |s| ≤ δ and a.e. x ∈ Ω  (3.4)

Thus, from (3.3) and (3.4) we deduce

F (x, s) ≤ \frac{β′}{2} |s|^{2} + A|s|^{μ} for |s| ≤ δ and a.e. x ∈ Ω  (3.5)

Let u ∈ H_{0}^{1}(Ω), via (3.5) and using the Poincaré inequality λ_{1} ∫ u^{2} ≤ ∥u∥^{2} and the Sobolev inequality ∫ u^{μ} ≤ K∥u∥^{μ}, we obtain

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

≥ \frac{λ_{1} − λ_{k}}{2} ∫ u^{2} − \frac{β′}{2} ∫ u^{2} − A ∫ u^{μ}

≥ \frac{λ_{1} − λ_{k} − β′}{2λ_{1}} ∥u∥^{2} − AK∥u∥^{μ}

Since μ > 2, we obtain the estimates

Φ(u) ≥ (\frac{λ_{1} − λ_{k} − β′}{2λ_{1}} − AK∥u∥^{μ−2})∥u∥^{2} ≥ \frac{λ_{1} − λ_{k} − β′}{4λ_{1}} r^{2} = ν ≥ 0 ∀∥u∥ = r

with r = (\frac{λ_{1}−λ_{k}−β′}{4λ_{1}AK})^{\frac{1}{μ−2}} , wich proves the lemma. □

Proof of Theorem 1.1 The first assertion of lemma 3.2 implies that there exists v ∈ H_{0}^{1} such that Φ(tv) → −∞ as t → ∞. In view of lemmas 3.1 and 3.2, we may apply Theorem 2.1 letting S = {u | ∥u∥ = r} and Q = {tv | v ∈ E_{−} ⊕ E_{k}, 0 ≤ t ≤ t_{0}}, with t_{0} > 0 being such that Φ(tv) ≤ 0. It follows that the functional Φ has a critical value c ≥ β > 0 and, hence, problem (1.1) has a nontrivial solution u ∈ H_{0}^{1}.

4 Existence of multiple nontrivial solutions

In this section, we consider the existence of multiple nontrivial solutions of problem (1.1). It is well known That, under the conditions of f in Theorem 1.2 or Theorem 1.4, Φ is a C^{2} functional with

Φ′(u)v = ∫_{Ω} ∇u∇v − λ_{k} ∫ uv − ∫ f(x, u)v, for u, v ∈ H_{0}^{1}(Ω),

Φ′′(u).v.w = ∫_{Ω} ∇w∇v − λ_{k} ∫ wv − ∫ f′(x, u)wv, for u, w, v ∈ H_{0}^{1}(Ω).

Set, Φ^{c} = {u ∈ H_{0}^{1}(Ω) | Φ(u) ≤ c}. 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 = 0, 1, 2, …,

where U is a closed neighborhood of u.

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.

Now, we will prove the following lemmas.

Lemma 4.1 There is an r > 0 and an α > 0 such that Φ ≥ α on ∂B_{r}(0).

Proof: Using the Poincare’s inequality, we have for every v ∈ H_{0}^{1}(Ω),

Φ′′(0).v.v = ∫_{Ω} |∇v|^{2} − ∫ [λ_{k} + f′(x, 0)]v^{2} dx.

Put m(x) = λ_{k} + f′(x, 0), and so in the case where m(x) ≤ 0, then

Φ′′(0).v.v ≥ ∥v∥^{2}.

In the case where m(x) > 0 on subset of positive measure we have λ_{1}(m) > 1. It follows from the Poincaré’s inequality that

Φ′′(0).v.v ≥ (1 − \frac{1}{λ_{1}(m)}∥v∥^{2}.

It follows that 0 is a non-degenerate critical point of Φ with the Morse Index of Φ at 0 is 0, and so it is well known that C_{q}(Φ, 0) = δ_{q0}ℤ.

For sufficiently small ρ > 0 we have for ∥u∥ ≤ ρ

Φ(u) = \frac{1}{2}Φ”(0).u.u + o(∥u∥^{2}) ≤ \frac{1}{2}[\frac{Φ”(0).u.u + o(∥u∥^{2})}{∥u∥^{2}}]∥u∥^{2}

with \frac{o(∥u∥^{2})}{∥u∥^{2}} → 0 as ∥u∥ → 0. For r < ρ, there holds

Φ(u) ≥ \frac{1}{4}(1 − ξ(m))r^{2}, ∥u∥ = r

with

ξ(m) = \begin{cases}\frac{1}{λ_{1}(m)} & if m > 0 on subset of positive measure \\ 0 & if m ≤ 0 \end{cases}

Since λ_{1}(m) > 1, clearly there exists α > 0 such that Φ(u) ≥ α on ∂B_{r}(0).

Lemma 4.2 There exists at least one nontrivial critical point u_{0} of Φ such that C_{q}(Φ, u_{0}) = δ_{q1}ℤ.

Proof: It easy to see that the conditions (F4), (F5) follow from conditions (F2), (F3). Then, by lemma 3.2, Φ is anticoercive on E^{−} ⊕ E^{k} we can find a w such that ∥w∥ > R and Φ(w) ≤ 0. The compactness condition and geometry of Φ assure that c = inf_{h∈Γ} max_{0≤t≤1} Φ(h(t)) is a critical value for Φ, where Γ = {h ∈ C([0, 1], H_{0}^{1}) | h(0) = 0, h(1) = w}, and c ≥ α. Then, there exists a critical point nontrivial, u_{0}, of mountain pass type such that C_{1}(Φ, u_{0}) ≠ 0.

If u_{0} is a non-degenerate critical point with its Morse index is k and then C_{q}(Φ, u_{0}) = δ_{qk}ℤ. Since C_{1}(Φ, u_{0}) ≠ 0 it follows that C_{q}(Φ, u_{0}) = δ_{q1}ℤ.

If u_{0} is degenerate, by Gromoll-Meyers theorem (cf. [16]), we have

C_{q}(Φ, u_{0}) = 0 for q < k and q > j + k

with j = dim KerΦ′′(u_{0}). Since C_{1}(Φ, u_{0}) ≠ 0, it follows that k ≤ 1.

If k = 1, by the shifting theorem and the critical group characterization of the local minimum, we have C_{q}(Φ, u_{0}) = δ_{q1}ℤ.

If k = 0, we have

Φ′′(0).v.v = ∫_{Ω} |∇v|^{2} − ∫ [λ_{k} + f′(x, 0)]v^{2} dx ≥ 0.

for every v ∈ H_{0}^{1}(Ω) and according to a result of Manes-Micheletti (cf. [15]), we have j = dim KerΦ′′(u_{0}) = 1 . Consequently, from the shifting theorem and the critical group characterization of the local maximum, we have C_{q}(Φ, u_{0}) = δ_{q1}ℤ.

Proof of Theorem 1.2 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_{k}u+f(x, 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. In a similar way we can define a compact map T : H_{0}^{1}(Ω) → H_{0}^{1}(Ω) by < T u, v >= ∫ (λ_{k} + μ)uv. It is well know for 0 < μ < λ_{k+1} − λ_{k} that

deg(I − T, B_{R}(0), 0) = (−1)^{m}

where m represents the dimension of V^{−} ⊕ V^{k} and B_{R}(0) the ball in H_{0}^{1}(Ω) of radius R > 0. Suppose {0, u_{0}} is the critical set of Φ and let R > 0 such that {0, u_{0}} ⊂ B_{R}(0). According to [7, theorem 3.2, capt. II] and the addition property of Leray-Schauder degree imply

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})

= dim C_{0}(Φ, 0) + dim C_{1}(Φ, u_{0})

= 1 − 1 = 0.

But this contradicts the fact that

deg(Φ′, B_{R}(0), 0) = (−1)^{m}.

Indeed, in [17] Omari and Zanolin establish an a priori bound for the solution set of the family equation

−(Δu + λ_{k}u) = (1 − t)μu + tf(x, u), x ∈ Ω, t ∈ [0, 1]
u = 0 on ∂Ω

with, 0 < μ < λ_{k+1} − λ_{k}. The homotopy invariance of Leray-Schauder degree implies that deg(Φ′, B_{R}(0), 0) = deg(I − T, B_{R}(0), 0) = (−1)^{m}.

Proof of corollary 1.1 ¿From (F8), there are s_{1}, s_{2} such that 0 < f′(s_{1}), f′(s_{2}) < λ_{k+1} − λ_{k} = μ. Thus, by continuity, there exist s_{3} ∈ ℝ, ε > 0, δ > 0 such that

ε ≤ f′(s) ≤ μ − ε

for every s ∈ [s_{3} − δ, s_{3} + δ]. Let s_{3} ≥ 0 be and take s > s_{3} + δ, we have

εδ ≤ f(s) = ∫_{0}^{s_{3}} f(t) dt + ∫_{s_{3}}^{s_{3}+δ} f(t) dt + ∫_{s_{3}+δ}^{s} f(t) dt ≤ μs − εδ.

Hence, the corollary follows. Similarly, we have F_{5}) and F_{6}), if s_{3} < 0. □ For the proof of Theorem 1.4, we will need the following lemma.

Lemma 4.3 The functional Φ is coercive in E(λ_{1})^{⊥} = W. Moreover, the functional Φ has at least nontrivial critical point of mountain pass type.

Proof. Suppose by contradiction that Φ is not coercive in W. Thus, there is some constant B and some sequence (w_{n}) ⊂ W, with ∥w_{n}∥ → ∞, such that

Φ(w_{n}) = \frac{1}{2} ∫ [|∇w_{n}|^{2} dx − λ_{k+1}w_{n}^{2}] + ∫ [\frac{1}{2}(λ_{k+1} − λ_{k})w_{n}^{2} − F (x, w_{n})] dx ≤ B

This implies that ∥w_{n}^{+}∥ is bounded, and so Lemma 2.3 gives us a contradiction from (F3). Therefore, Φ is bounded from below in W. Hence, since Φ is weakly lower semi-continuous and coercive on W, Φ attains the infimum β = inf_{W} Φ.

On the other hand, since Φ is anticoercive on E(λ_{1}), we can find t_{0} > 0 that Φ(±t_{0}φ_{1}) < β. In view of Lemma 3.1, we may apply Theorem 2.1 to get that

c = inf_{h∈Γ} max_{0≤t≤1} Φ(h(t)) ≥ β

is a critical value for Φ, where Γ = {h ∈ C([0, 1], H_{0}^{1}) | h(0) = −t_{0}φ_{1}, h(1) = t_{0}φ_{1}}. As in the proof of Theorem 1.2, there exists a critical point u_{0} of Φ such that C_{q}(Φ, u_{0}) = δ_{q1}ℤ.

Now we will prove that 0 is a nondegenerate critical point with Morse index of Φ at 0 equal d.

Claim. The Morse index of Φ at 0 is d, with d ≥ 2.

In fact, for every v ∈ ⊕_{j≤m}E^{j} we have

Φ′′(0).v.v = ∫_{Ω} |∇v|^{2} − ∫ [λ_{1} + f′(x, 0)]v^{2} dx ≤ ∫ λ_{m} − [λ_{1} + f′(x, 0)]v^{2} dx.

On the other hand, for every v ∈ ⊕_{j≥m+1}E^{j} we have

Φ′′(0).v.v ≥ ∫ λ_{m+1} − [λ_{1} + f′(x, 0)]v^{2} dx.

¿From (F8) we obtain that 0 is a nondegenerate critical point of Φ with Morse index is d, where d is the the dimension of ⊕_{j≤m}E^{j} and clearly d is larger than 2. The last claim implies that C_{q}(Φ, 0) = δ_{qd}ℤ, and so u_{0} is a nontrivial critical point of mountain pass type.

Proof of Theorem 1.4 As in the proof of Theorem 1.2, we assume {0, u_{0}} is the critical set of Φ. Let R > 0 such that {0, u_{0}} ⊂ B_{R}(0). According to [7, Theorem 3.2, Capt.II] and the addition property of Leray-Schauder degree imply

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})

= dim C_{0}(Φ, 0) + dim C_{1}(Φ, u_{0})

= (−1)^{d} − 1.

Observe that the right hand side of this equality is even, which contradicts the claim given in the proof of Theorem 1.2. Hence, problem (1.1) has at least two nontrivial solutions. □

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Abdel R. El Amrouss University Mohamed I, Faculty of sciences Department of Mathematics, Oujda; Morocco e-mail: amrouss@sciences.univ-oujda.ac.ma