On a Positive Solutions for (p, q)-Laplacian Steklov Problem with Two Parameters

3 Main results 5 3.1 Case (α, β) ∈ R\[λ1(p), +∞) × [λ1(q), +∞) . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Case (α, β) ∈ [λ1(p), +∞) × [λ1(q), +∞) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2.1 Instruction of the curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2.2 Existence and non-existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7


Introduction
In this paper, we prove various existence and non-existence of positive solutions for the following (p, q)-Laplacian Steklov eigenvalue problem: (P α,β ) div[A p,q (|∇u|)∇u] = A p,q (u)u in Ω, A p,q (|∇u|)∇u, ν = α|u| p−2 u + β|u| q−2 u on ∂Ω where Ω is a bounded domain in R N (N ≥ 2) with smooth boundary ∂Ω, ν is the outward unit normal vector on ∂Ω, ., . is the scalar product of R N . α, β ∈ R, A p,q (s) = |s| p−2 + |s| q−2 and 1 < q < p < ∞. Not that the assumption q < p is taken without loss of generality, due to the symmetry of symbols in (P α,β ); therefore all results of the present work have corresponding counterparts in the case p > q. It is easy to see that div[A p,q (∇u)] = ∆ p + ∆ q , called (p, q)-Laplacian, occurs in quantum field theory, where ∆ p = div(|∇u| p−2 ∇u).
The problem (P α,β ) comes, for example, from a general reaction diffusion system u t = div(D(u)∇u) + c(x, u), (1.1) where D(u) = (|∇u| p−2 + |∇u| q−2 ). This system has a wide range of applications in physics and related sciences like chemical reaction design [2], biophysics [8] and plasma physics [17]. In such applications, the function u describes a concentration, the first term on the right-hand side of (1.1) corresponds to the diffusion with a diffusion coefficient D(u); whereas the second one is the reaction and relates to source and loss processes. Typically, in chemical and biological applications, the reaction term c(x; u) has a polynomial form with respect to the concentration. In the last few years, the (p, q)-Laplace attracts a lot 2 A. Boukhsas, A. Zerouali, O. Chakrone and B. Karim of attention and has been studied by many authors (see [12,16,20,23]). However, there are few results one the eigenvalue problems for the (p, q)-Laplacian, we cite [3,7,14,18]. Under the zero Dirichlet boundary condition in Ω, the authors obtained in [15] reasonably complete description of the subsets of (α, β) plane which correspond to the existence/nonexistence of positive solutions to the following problem: in Ω, u = 0 on ∂Ω, (1.2) where m r ∈ L ∞ (Ω), m r ≡ 0 and m r ≥ 0 a.e in Ω for r = p, q. In [21,22] the authors of the present article studied the existence and non-existence results of a positive solution for the Steklov eigenvalue problem P (λ,λ) ( α = β = λ).
Our goal in this paper is to provide a complete description of 2-dimensional sets in the (α, β) plane, which correspond to the existence and non-existence of positive solutions for (P α,β ) by generalizing and complementing the research [21,22], and seems more natural, due to the structure of the equation. We restrict ourselves to the case where m p and m q are constants, to save transparency and simplicity of presentation. However, we emphasize that all the results of the present article remain valid for the following problem with non-negative weights (P α,β,mp,mq ): in Ω, |∇u| p−2 ∇u + |∇u| q−2 ∇u, ν = αm p (x)|u| p−2 u + βm q (x)|u| q−2 u on ∂Ω (1.3) Let r = p, q and let N −1 r−1 < s r < ∞ if r < N and s r ≥ 1 if r ≥ N . The function weight m r ∈ M + r , where M + r := {m r ∈ L sr (∂Ω); m r ≡ 0, m r ≥ 0}. Hereinafter, u 1,r := u W 1,r (Ω) denotes the norm of Sobolev space W 1,r (Ω). We say that u ∈ W 1,p (Ω) is a weak solution of (P α,β ) if its holds As usual, we say that λ is an eigenvalue of ∆ r with weight function m r ∈ M r if the problem has a non-trivial solution. If the Lebesgue measure of {x ∈ Ω : m r (x) > 0} is positive, then (P λ ) possesses the first positive eigenvalue λ 1 (r, m r ) (cf. [11]) that can be obtained by minimizing the Rayleigh quotient: where Φ(u) := Ω |∇u| r dx + Ω |u| r dx and Ψ(u) := ∂Ω m r |u| r dσ. Note that λ 1 (r, m r ) is simple and isolated. It is also worth mentioning that λ 1 (r, m r ) has positive eigenfunctions ϕ 1 (r, m r ) ∈ C 1,αr (Ω) with some α r ∈ (0, 1) (see [1]). Hereinafter we will also use the notation λ 1 (r) := λ 1 (r, 1) for the first eigenvalue of ∆ p without weight and ϕ r for the corresponding eigenfunction.
In what follows, we will say that λ 1 (p) and λ 1 (q) have different eigenspaces if the corresponding eigenfunctions ϕ p and ϕ q are linearly independent, i.e. the following assumption is satisfied: Let us note that availability or violation of the assumption (1.5) significantly affects the sets of existence of solutions for P (α,β) , see Fig. 1 and the section 3 for precise statements. The rest of this paper is organized as follows. In section 2, we give some preliminary results and definitions which are needed in the proof of the main results. Section 3, we state our main results. Section 4, we prove the existence of solution for P (α,β) in the neighborhood of (λ 1 (p), λ 1 (q)) provided (1.5) is satisfied. Section 5, we prove our results stated in section 4.

Preliminaries
In this section, we give some preliminary results and definitions which well be used in the following sections. First, we give three results from [21,22], where they were proved using the variational methods. Theorem 2.1. ( [21], Theorem 2.5) One assumes that m p ∈ M p and m q ∈ M q . If 0 < λ < min{λ 1 (p, m p ), λ 1 (q, m q )}, then the problem (P λ,λ,mp,mq ) has no non-trivial solutions. Theorem 2.2. ( [21], Theorem 3.1) One supposes that m p ∈ M p , m q ∈ M q and λ 1 (p, m p ) = λ 1 (q, m q ). If then the problem (P λ,λ,mp,mq ) has at least one positive solution. and then the problem (P λ,λ,mp,mq ) has at least one positive solution.
Next, we introduce the super-and sub-solution method for the problem (P α,β ). We recall the notations of sub-and super-solutions of our problem.
Recalling that q < p and β − λ 1 (q) > 0, we see that E α,β [0,u] (tϕ q ) < 0 for sufficiently small t > 0, whence u] (tϕ q has a non-trivial critical point, and our conclusion follows. Finally, we give the Picone's identity for (p, q)-Laplacian. We state a proposition that well be used.

Main results
In this section we state our main results.

Case
First, we state the result of non-existence which generalize Theorem 2.1 from [21] for the problem (P α,β , m p , m q ) with non-negative weights.

Case
3.2.1. Instruction of the curve. Note first that for any α, β ∈ R the problem (P α,β ) is equivalent to (P β+s,β ), where s = α − β. Denoting now, for convenience, λ = β, for each s ∈ R we consider λ * (s) := sup{λ ∈ R : (P λ+s,λ ) has a positive solution}, (3.2) provided (P λ+s,λ ) has a positive solution for some λ. If there are no such λ, we set λ * (s) = −∞. Define also Notice that it is still unknown if there is s * − ∈ R, such that λ * (s) + s = λ 1 (p) for all s ≤ s * − , or λ * (s) + s > λ 1 (p) for all s ∈ R, whenever (1.5) is satisfied. Now we define the curve C in(α, β) plane as follows: From proposition 3.4 there directly follow the corresponding conclusion for C, namely, C is locally finite, . We especially note that λ 1 (s) + s = λ 1 (p) for s ≤ s * and λ 1 (s) = λ 1 (q) for s ≥ s * if and only if (1.5) doesn't hold. It directly follows from the combination of the criterion (iv), estimations (ii) and monotonicity (vi) from Proposition 3.4. In other words, our curve C coincides with the polygonal line Let us note that the main disadvantage of characterization 3.2 of λ * (s) is its non-constructive form. However, using the extended functional method (see [5,10]) we provide the equivalent characterization of λ * (s) by an explicit minimax formula, which can be used in further numerical investigations of (P α,β ): The next proposition shows that (3.2) and (3.3) are in fact, equivalent.

Existence and non-existence results.
First, we state our second main theorem, which shows that (P α,β ) possesses a positive solution if (α, β) is below the curve C, and has no positive solutions if (α, β) is above C.
Next, we provide the results about existence and non-existence on the curve C.

Existence result in the neighborhood of
In this section we prove the existence of solution for α = λ 1 (p)+ ε, β = λ 1 (q)+ ε under the assumption (1.5). First, we define the energy functional corresponding to (P α,β ) by where for further simplicity we denote Note that E α,β ∈ C 1 (W 1,p (Ω), R). Next, we introduce the so-called Nehari manifold (see [4]) Proof. Fix some non-trivial function u ∈ W 1,p (Ω) and consider the fibred functional corresponding to E α,β (u): is satisfied for unique t > 0 given by This implies that and hence t(u)u ∈ N α,β . Moreover, recalling that q < p, if (4.1) holds, then

Proofs of main results
In this section, we collect the proofs of our results stated in section 3.
Proof. Proof of Proposition 3.1. Let α ≤ λ 1 (p) and β ≤ λ 1 (q). Assume that u ∈ W 1,p (Ω) is a non-trivial solution of (P α,β ). Taking u as a test function we have This chain of inequalities is satisfied if and only if α = λ 1 (p), β = λ 1 (q) and u is the eigenfunction corresponding to λ 1 (p) and λ 1 (q) simultaneously. As a result, our conclusion is shown.

Proof of Proposition 3.4.
Here we prove Properties of λ * (s) Part (i). Fix any s ∈ R and let u ∈ W 1,p (Ω) be a positive solution of (P λ+s,λ ) for some λ ∈ R.
Proof. Proof of Theorem 3.3. Note first that from Proposition 3.1 and 3.2 it directly follows that if (3.1) is satisfied, then (P α,β ) has at least one positive solution.