Multiple solutions for p-Laplacian eigenproblem with nonlinear boundary conditions

abstract: In this paper we study the existence of at least two nontrivial solutions for the nonlinear p-Laplacian problem, with nonlinear boundary conditions. We establish that there exist at least two solutions, which are opposite signs. For this reason, we characterize the first eigenvalue of an intermediary eigenvalue problem by the minimization method. In fact, in some sense, we establish the non-resonance below the first eigenvalues of nonlinear Steklov-Robin problem.


Introduction
Let us consider the following nonlinear boundary problem where Ω ⊂ R N , N ≥ 1, be a bounded domain with smooth boundary ∂Ω, the functions c : Ω → R, f : Ω × R → R and g : ∂Ω × R → R satisfy the following conditions: (C 1 ) c ∈ L p (Ω) and c ≥ 0 a.e x ∈ Ω with Ω c(x)dx > 0; (C 2 ) f and g are Carathéodory functions; The nonlinear p-Laplacian problem with Dirichlet, Neumann or Stecklov conditions has been studied by severals authors, for example, we cite the papers [1,7] and [2] in which the authors established the existence of positive and multiple solutions for the following quasilinear problem on ∂Ω, under appropriate assumptions on f .In this sense we have extended this work to a nonlinear problem which has a strongly nonlinear second member.The resolution of the problem (S 1 ) appears in several works, we cite for example [6,8], but they are only in the case of quasilinear problems.This paper is organized as follows.
In Section 2, we give a characterization of the first eigenvalue of the problem (S 2 ) bellow.In Section 3, we give some preliminary results and notations, that will be useful to prove the principal results of this article.In Section 4, we establish the main result.

Characterization of the first eigenvalues
Firstly, we consider the following eigenvalue problem The variational problem associated to the problem (S 2 ), see [5], is given by 1) The existence of the eigenvalues sequence (λ k ) k≥1 of the problem (S 2 ), see [4] and [5], is given as follows where , compact, symmetric and γ(B) ≥ k} , γ is the genus's function and 2) is given by the L.Ljusternik-L.Schnirelmann critical point theory on C 1 manifolds using the genus γ.The first eigenvalue is characterized by We have proved in [5] that the first eigenvalue λ 1 is simple and an eigenfunction u 1 associated with it, does not change sign in Ω and |u 1 | > 0 in Ω.
Definition 2.1.A weak solution of problem (S 1 ) is a function u ∈ W 1,p (Ω) satisfied

Preliminary results
We consider the following truncated problem We denote by u + = max(u, 0) and u − = max(−u, 0).We also consider the functions F ± and G ± are defined as follows We assume the following assumptions and Lemma 3.1.All solutions for (S 1+ ) ( respectively (S 1− )) are the positive (respectively negative) solutions for (S 1 ).
Let u be a solution of (S 1+ ), then it is clear that u is a critical point of φ + .We have for all v ∈ W 1,p (Ω), We take v = u − in (3.5), we have which proves that u − = 0, so u = u + , is also a critical point of φ, with the critical value is φ(u) = φ(u + ) = φ + (u).Similarly for the problem (S 1− ).✷ We pose

Main result
Theorem 4.1.Suppose the conditions (C 2 ) − (C 4 ) and or Then, the nonlinear problem (S 1 ), has at least two (non trivial) solutions in W 1,p (Ω).where in the first case is positive and the second is negative.
For proof of theorem, we need the following lemma Proof: Let (u n ) n a sequence satisfying the (P.S) conditions, (φ(u n ) n ) n is bounded and φ ′ (u n ) → 0. We show that (u n ) n has a convergent subsequence.As W 1,p (Ω) ֒→ Multiple solutions for p-Laplacian eigenproblem

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L p (Ω), it suffices to show that (u n ) n is a bounded sequence.By absurd, pose v n = un un , with u n → +∞, so v n = 1.As (v n ) n is bounded in W 1,p (Ω), then there exist a subsequence also noted (v n ) n such that, Firstly, we show that v = 0.By absurd, we suppose that v = 0, so so (4.5) As f+(x,un) |un| p−2 un and g+(x,un) |un| p−2 un are bounded in L p ′ (Ω) and L p ′ (∂Ω) respectively, v n → 0 in L p (Ω) and v n → 0 in L p (∂Ω), we deduce that Therefore, we have 0 = 1 − 0 − 0, it's absurd.Consequently v = 0. Now we prove that (u n ) is bounded.

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The same, we prove that As .
Now, we prove that φ + satisfies the geometric conditions of the mountain pass, in order to prove that φ + admits a critical point.We have For sufficiently small ǫ, we have 1 − ≤ 0, ( because λ 1 (k + , l + ) < 1).
Multiple solutions for p-Laplacian eigenproblem
For w = u − and as for all t ≤ 0, f + (x, t) = 0 and g + (x, t) = 0, we obtain Thus u − = 0, so u − = 0. Which proves that u > 0 is the solution of the problem (S 1 ).