A class of equations with three solutions

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta u=\lambda u&in $\Omega$ \cr&\cr u=0&on $\partial\Omega$.\cr}$$ Then, for every $\lambda>\lambda_1$ and for every convex set $S\subseteq H^1_0(\Omega)$ dense in $H^1_0(\Omega)$, there exists $\alpha\in S$ such that the problem $$\cases{-\Delta u=\lambda(u^+-(u^+)^q)+\alpha(x)&in $\Omega$ \cr&\cr u=0&on $\partial\Omega$\cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(\Omega)$ of the functional $$u\to {{1}\over {2}}\int_{\Omega}|\nabla u(x)|^2dx-\lambda\int_{\Omega}\left ({{1}\over {2}}|u^+(x)|^2-{{1}\over {q+1}}|u^+(x)|^{q+1}\right )dx-\int_{\Omega}\alpha(x)u(x)dx\ $$ where $u^+=\max\{u,0\}$.


Introduction
There is no doubt that the study of nonlinear PDEs lies in the core of Nonlinear Analysis. In turn, one of the most studied topics concerning nonlinear PDEs is the multiplicity of solutions. On the other hand, the study of the global minima of integral functionals is essentially the central subject of the Calculus of Variations. In the light of these facts, it is hardly understable why the number of the known results on multiple global minima of integral functionals is extremely low. Certainly, this is not due to a lack of intrinsic mathematical interest. Probably, the reason could reside in the fact that there is not an abstract tool which has the same popularity as the one that, for instance, the Lyusternik-Schnirelmann theory and the Morse theory have in dealing with multiple solutions for nonlinear PDEs.
Abstract results on the multiplicity of global minima, however, are already present in the literature. We mainly allude to the result first obtained in [1] and then extended in [2] and [5] which ensures the existence of at least two global minima provided that a strict minimax inequality holds. We already have obtained a variety of applications upon different ways of checking the required strict inequality ( [3], [4], [6]).
The aim of the present paper is to establish an application of Theorem 1 of [7] which is itself an application of the main result in [5]. Precisely, we first establish a general result which ensures the existence of three solutions for a certain equation provided that another related one has no non-zero solutions (Theorem 1). Then, we present an application to nonlinear elliptic equations (Theorem 2).
We say that I is coercive if lim x X →+∞ I(x) = +∞. We also say that I ′ admits a continuous inverse on X * if there exists a continuous operator T : X * → X such that T (I ′ (x)) = x for all x ∈ X.
Here is our abstract result: THEOREM 1. Let I be weakly lower semicontinuous and coercive, and let I ′ admit a continuous inverse on X * . Moreover, assume that the operators ϕ ′ and ψ ′ are compact and that Set and assume that θ * <θ .
Then, for each λ ∈]θ * ,θ[, with λ ≥ 0, either the equation has a non-zero solution, or, for each convex set S ⊆ Y dense in Y , there existsỹ ∈ S such that the equation has at least three solutions, two of which are global minima in X of the functional As it was said in the Introduction, the main tool to prove Theorem 1 is a result recently obtained in [7]. For reader's convenience, we now recall its statement: , Theorem 1). -Let X, E be two real reflexive Banach spaces and let Φ : X × E → R be a C 1 functional satisfying the following conditions: there exists a convex set S ⊆ E dense in E, such that, for each y ∈ S, the functional Φ(·, y) is weakly lower semicontinuous, coercive and satisfies the Palais-Smale condition .
Then, either the system has a solution (x * , y * ) such that or, for every convex set T ⊆ S dense in E, there existsỹ ∈ T such that equation has at least three solutions, two of which are global minima in X of the functional Φ(·,ỹ).
Proof of Theorem 1. Fix λ ∈]θ * ,θ[, with λ ≥ 0. Assume that the equation has no non-zero solution. Fix a convex set S ⊆ Y dense in Y . We have to show that there existsỹ ∈ S such that the equation has at least three solutions, two of which are global minima in X of the functional To this end, let us apply Theorem A. Consider the functional Φ : On the other hand, we have and so the sequence {∂ x ϕ(x n ), y (·)} converges in X * to η(·)(y). Then, since ψ ′ is compact, the operator ∂ x ϕ(·), y +λψ ′ (·) is compact too. From this, it follows that ϕ(·), y +λψ(·) is sequentially weakly continuous ( [8], Corollary 41.9). If x X is large enough, we have I(x) > 0 and so we can write In view of (1), we also have lim inf We claim that lim sup .
and so (5) is satisfied just since λ <θ. Since I is coercive and weakly lower semicontinuous, the functional Φ(·, y) turns out to be coercive, in view of (2), (3), (4), and weakly lower semicontinuous, in view of the Eberlein-Smulyan theorem. Finally, since I ′ admits a continuous inverse on X * , Φ(·, y) satisfies the Palais-Smale condition in view of Example 38.25 of [8]. Hence, Φ satisfies the assumptions of Theorem A. Now, we claim that there is no solution (x * , y * ) of the system Arguing by contradiction, assume that such a (x * , y * ) does exist. This amounts to say that and Therefore So, by the initial assumption, we have x * = 0 and hence y * = 0 (recall that ϕ(0) = 0). As a consequence, since I(0) = ψ(0) = 0, (6) becomes inf x∈X (I(x) − λψ(x)) = 0 .
Now, notice that (7) contradicts the fact that λ > θ * . Hence, a fortiori, the system and then the existence ofỹ ∈ S is directly ensured by Theorem A. △ We now present an application of Theorem 1 to a class of nonlinear elliptic equations.
Let Ω ⊂ R n be a smooth bounded domain. We denote by A the class of all Carathéodory's functions f : Ω × R → R such that, for each u, v ∈ H 1 0 (Ω), the function x → f (x, u(x))v(x) lies in L 1 (Ω). For f ∈ A, we consider the Dirichlet problem As usual, a weak solution of the problem is any u ∈ H 1 0 (Ω) such that for all v ∈ H 1 0 (Ω). Also, we denote by λ 1 the first eigenvalue of the Dirichlet problem For any continuous function f : R → R, we set F (ξ) = ξ 0 f (t)dt for all ξ ∈ R. THEOREM 2. -Let f, g : R → R be two continuous functions satisfying the following growth conditions: and assume that max{ρ, 0} < σ .
Then, for every λ ∈ λ1 2σ , λ1 2 max{ρ,0} (with the conventions λ1 +∞ = 0, λ1 0 = +∞), either the problem has a non-zero weak solution, or, for every convex set S ⊆ H 1 0 (Ω) dense in H 1 0 (Ω), there exists α ∈ S such that the problem has at least three weak solutions, two of which are global minima in H 1 0 (Ω) of the functional PROOF. We are going to apply Theorem 1 taking X = H 1 0 (Ω), Y = L 2 (Ω), with their usual scalar products (that is, u, v X = Ω ∇u(x)∇v(x)dx and u, v Y = Ω u(x)v(x)dx), and for all u ∈ X. In view of (b), thanks to the Sobolev embedding theorem, the operator ϕ and the functional ψ are C 1 , with compact derivative. Moreover, the solutions of the equation are weak solutions of (8) and, for each α ∈ Y , the solutions of the equation are weak solutions of (9). Moreover, condition (1) follows readily from (a) which is automatically satisfied when n ≥ 4 since p < 2 n−2 . We claim that lim sup Indeed, fix ν > ρ. Then, there exists δ > 0 such that G and so lim sup Now, we get (10) passing in (12) to the limit for ν tending to ρ. We also claim that Indeed, fix η < σ. For instance, let σ = lim inf ξ→0 + G(ξ) ξ 2 . Then, there exists η > 0 such that Now, (13) is obtained from (15) passing to the limit for η tending to σ. Now, fix λ ∈ λ1 2σ , λ1 2 max{ρ,0} . Then, from (10) and (13), we obtain lim sup This readily implies that θ * < λ <θ and the conclusion is directly provided by Theorem 1. △ COROLLARY 1. -Let the assumptions of Theorem 2 be satisfied and let λ ∈ λ1 2σ , λ1 2 max{ρ,0} satisfy Then, for every convex set S ⊆ H 1 0 (Ω) dense in H 1 0 (Ω), there exists α ∈ S such that the problem on ∂Ω has at least three weak solutions, two of which are global minima in H 1 0 (Ω) of the functional PROOF. It suffices to observe that, in view of (16), 0 is the only solution of (8) and then to apply Theorem 2. △ Finally, notice the following remarkable corollary of Corollary 1: Of course (with the notations of Theorem 2), ρ = 0 and σ = 1 2 . Since f in non-negative, F f is so in [0, +∞[ and non-positive in ] − ∞, 0]. Therefore, (16) is satisfied for all ξ ∈ R \ [0, 1] since g has the opposite sign of F f in that set. Now, let ξ ∈]0, 1]. We have which gives (16). Now, let S be any convex set S ⊆ H 1 0 (Ω) dense in H 1 0 (Ω). Then, the set γ λ S is convex and dense in H 1 0 (Ω) and the conclusion follows applying Corollary 1 with this set. △