On a Positive Solution for ( p , q )-Laplace Equation with Nonlinear Boundary Conditions and Indefinite Weights

abstract: In the present paper, we study the existence and non-existence results of a positive solution for the Steklov eigenvalue problem driven by nonhomogeneous operator (p, q)-Laplacian with indefinite weights. We also prove, under appropriate conditions, that the results are completely different from those for the usual Steklov eigenvalue problem involving the p-Laplacian with indefinite weight. Precisely, we show that there exists an interval of principal eigenvalues for our Steklov eigenvalue problem.


Introduction
Consider the (p, q)-Laplacian Steklov eigenvalue problem p,q (∇u)] = A (µ) p,q (u) in Ω, A (µ) p,q (∇u), ν = λ[m p (x)|u| p−2 u + µm q (x)|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 , λ ∈ R, µ ≥ 0 and 1 < q < p < ∞.Let r = p, q and let N −1 r−1 < s r < ∞ if r < N and s r ≥ 1 if r ≥ N .A (µ) p,q (s) = |s| p−2 s + µ|s| q−2 s and the function weight m r ∈ M r may be unbounded and change sign, where M r := {m r ∈ L sr (∂Ω); m + r ≡ 0}.The problem (P λ,µ ) 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 + µ|∇u| q−2 ).This system has a wide range of applications in physics and related sciences like chemical reaction design [2], biophysics [5] and plasma physics [14].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.The nonhomogeneous operator (p, q)-Laplacian have been the topic of many studies (see [6,7,13,17]).However, there are few results one the eigenvalue problems for the (p, q)-Laplacian, we cite [3,9,10,15].The classical eigenvalue problem for the (p, q)-Laplacian where △ r u = div (|∇u| r−2 ∇u) indicate the r-Laplacian, has attracted considerable attention.In [12], the authors study the problem (1.2) for domains with boundary C 2 and bounded weights.They proved, in the case where µ > 0, the existence of an interval of eigenvalues and the existence of positive solutions in nonresonant cases.A non-existence result is also given.In [18], A. Zerouali and B. Karim are proved the same results by assuming the singularities on the domain and the weights.Our purpose in this article is to extend the results of the classical eigenvalue problem involving the (p, q)-Laplacian (see for example [11,12]) and generalize some results knouwn in the classical p-Laplacian Steklov problems (see [4]).
We will write u r := Ω |u| r dx 1/r for the L r (Ω)−norm and W 1,r (Ω) will denote the usual Sobolev space with usual norm u W 1,r (Ω) := ( ∇u r r + u r r ) 1/r .We recall that a value λ ∈ R is an eigenvalue of problem (P λ,µ ) if and only if there exists u ∈ W 1,p (Ω)\{0} such that 3) for all ϕ ∈ W 1,p (Ω), where dσ is the N − 1 dimensional Hausdorff measure and u is then called an eigenfunction of λ.

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Nonlinear Steklov eigenvalue problem (P λ,mr ), where r = p, q and with indefinite weight m r ∈ M r have been studied by several authors, for example (see [4]).These works proved that there exists a first eigenvalue λ 1 (r, m r ) > 0, where which is simple in the sense that two eigenfunctions corresponding to it are proportional.Moreover, the corresponding first eigenfunction φ 1 (r, m r ) can be assumed to be positive.It was also shown in [4] that λ 1 (r, m r ) is isolated and monotone.This paper is divided into three sections, organized as follows.In Section 2, we study Rayleigh quotient for our problem (P λ,µ ).In contrast to homogeneous case, we prove that if λ 1 (p, m p ) = λ 1 (q, m q ) or φ 1 (p, m p ) = kφ 1 (q, m q ) for every k > 0, then the infimum in Rayleigh quotient is not attained.We also show non-existence results for positive solutions of the eigenvalue problem (P λ,µ ) formulated as Theorem 2.5.Our existence results for positive solutions of the eigenvalue problem (P λ,µ ) are presented in Section 3. We study the non-resonant case (Theorem 3.1) which prove that when µ > 0 there exists an interval of positive eigenvalues for the problem (P λ,µ ).

Rayleigh quotient and non-existence results
This section concerns the Rayleigh quotient and non-existence results for our eigenvalue Steklov problem (P λ,µ ).It is inspired from [11] and [15].Remark 2.1.We start by pointing out to find a solution for the problem (P λ,µ ) is equivalent to seek a solution in the case µ = 1, that is to solve the problem (P λ,1 ).Indeed, if u is a solution of (P λ,1 ), then multiplying equation (P λ,1 ) by s p−1 for s > 0 we deduce that v = su is a solution for problem (P λ,µ=s p−q ).Conversely, let u be a solution of problem (P λ,µ ).Then it follows that v = µ 1/p−q u is a solution of (P λ,1 ).

Rayleigh quotient for the problem (P λ,µ )
We introduce now the functionals A and B on W 1,p (Ω) by (2.1) for all u ∈ W 1,p (Ω).
Proposition 2.2.(i) The functional A is well defined and sequently weakly lower semi-continuous.
(ii) If m p ∈ M p and m q ∈ M q , then the functional B is also well defined and weakly continuous.
Proof.(i) The functional A is well defined.Indeed, since Ω bounded and q < p, we have W 1,p (Ω) ⊂ W 1,q (Ω).Then for all u ∈ W 1,p (Ω), 1 p u p W 1,p (Ω) < ∞ and The functional B is also well defined.Indeed, for u ∈ W 1,p (Ω), by Hölder's inequality, for r = p, q and s ′ r = s r /(s r − 1), we obtain Hence by Hölder's inequality, we have Thus the functional B is weakly continuous.✷ Define now the Rayleigh quotient For the proof of Proposition 2.3, we will need to use the following lemma.
Case (ii): Suppose that ∂Ω m p |u| p dσ ≤ 0 and ∂Ω m q |u| q dσ > 0. Using the definition of λ 1 (q, m q ), we also arrive at contradiction Case (iii): Suppose now that ∂Ω m p |u| p dσ > 0 and ∂Ω m q |u| q dσ > 0. It follows from the definition of λ 1 (r, m r ), where r = p, q that Hence we get (2.6) Against the assumption in our reasoning by contradiction.✷ Proof of Proposition 2.3.By contradiction, we suppose that: Using Lemma 2.4, we give We argue by considering the three cases in the proof of Lemma 2.4.Case (i): By (2.4), (2.7) and ∂Ω m q |u| q dσ ≤ 0, we have We deduce that ∂Ω m p |u| p dσ and u W 1,q (Ω) = 0.

Non-existence results
The following theorem is the main result of this section.
Theorem 2.5.One assumes that m p ∈ M p and m q ∈ M q .
(b) Moreover, if one of the following conditions holds then the problem (P λ,1 ), with λ = λ * has no non-trivial solutions.
Proof of Theorem 2.5.Assume by contradiction that there exists a non-trivial solution u of problem (P λ,1 ).Then, for every s > 0, we have that v = su is a non-trivial solution of problem (P λ,s p−q ) (see Remark 2.1).Choose s p−q = p/q and then act with su as test function on the problem (P λ,s p−q ).We arrive at 0 < pA(su) = pλB(su). (2.8) From the estimate (2.8) and according to Lemma 2.4, we obtain This contradiction yields the first assertion of the theorem.
The second part of the Theorem 2.5 follows by Proposition 2.3.✷
Remark 3.2.The proof of Theorem 3.1 reduces to provide a non-trivial critical point of the functional I λ,mp,mq defined for all u ∈ W 1,p (Ω) by where u + = max{u, 0} and A, B are the functionals defined by (2.1) and (2.2).This non-trivial critical point u of I λ,mp,mq is a non-negative solution of the problem (P λ,1 ).Indeed, inserting −u − = − max{−u, 0} as test function leads to thus u − = 0. We can check that u ∈ C 1,α (Ω) for some α ∈ (0, 1) (see [1]).Then the maximum principle of Vasquez [16] can be applied to ensure positiveness of u.
The argument will be separately developed in two cases: In case (a), we apply the minimum principle and in case (b), we use the mountain pass theorem.
Proof of case (a).By Proposition 2.2, A is sequently weakly lower semicontinuous and B is weakly continuous.It follows that I λ,mp,mq is sequently weakly lower semi-continuous.Moreover I λ,mp,mq is bounded from below.Indeed for all u ∈ W 1,p (Ω), we have It is remains to show that I λ,mp,mq is coercive in which is possible due to the assumption in case (a).For every u ∈ W 1,p (Ω) with ∂Ω m p (u + ) p dσ ≤ 0, through Holder's inequality we obtain for u ∈ W 1,p (Ω) with ∂Ω m p (u + ) p dσ > 0, by (1.4) we have On a Positive Solution for (p, q)-Laplace Equation . . .
Proof of case (b).We organize the proof of this case in several lemmas.In the sequel, we design by o(1) a quantity tending to 0 as n −→ ∞.Proof.Let (u n ) ⊂ W 1,p (Ω) be a sequence such that I λ,mp,mq (u n ) −→ c for c ∈ R and I ′ λ,mp,mq (u n ) −→ 0 in (W 1,p (Ω)) * as n −→ ∞.Let us first show that the sequence (u n ) is bounded in W 1,p (Ω).It is sufficient only to prove the boundedness of u n ps ′ p , because using the Hölder's inequality and the continuous embedding W 1,p (Ω) ⊂ L qs ′ q (∂Ω), we have where c ′ and c ′′ are the positive constants.Suppose by contradiction that u n ps ′ p → +∞ and let v n := where the positive contants C 1 and C 2 are defined by C 1 = c ′ λ m p sp and C 2 = c ′′ pλ q m q sq .Since 1 < q < p, the inequality (3.6) implies the boundedness of v n in W 1,p (Ω).for a subsequence, v n ⇀ v (weakly) in in W 1,p (Ω).By the compact embedding.W 1,r (Ω) ⊂ L rs ′ r (∂Ω), (r = p, q) we have v n → v strongly in L rs ′ r (∂Ω) (r = p, q).First we observe that v − ≡ 0 in Ω.In fact, acting with −u − n as test function, we have a contradiction since we assumed λ = λ 1 (p, m p ). Hence u n is bounded in W 1,p (Ω).For a subsequence, u n ⇀ u (weakly) in W 1,p (Ω) and u n → u (strongly) in L ps ′ p (∂Ω).We claim now that u n → u in W 1,p (Ω).As W 1,p (Ω) is reflexive and uniformly convex, it suffices to prove that u n W 1,p (Ω) → u W 1,p (Ω) .It is clear that (3.10) Using Hölder's inequality and for (r = p, q), we have Moreover, (3.10) and (3.11) imply that u n W 1,p (Ω) → u W 1,p (Ω) .Thus u n → u strongly in W 1,p (Ω).✷ Lemma 3.4.Let m p ∈ M p , m q ∈ M q .If λ < λ 1 (q, m q ), then there exist δ > 0 and ρ > 0 such that To prove the Lemma 3.4, we need the following lemma.Proof.By way contradiction, we assume that ; hence v n bounded in W 1,p (Ω), then there exists a subsequence that we still denote v n such that v n ⇀ v weakly in W 1,p (Ω).By the compact embedding W 1,r (Ω) ⊂ L rs ′ r (∂Ω)(r = p, q) we have v n → v strongly in L rs ′ r (∂Ω).By (3.14), we have v n L qs ′ q (∂Ω) < 1 n , thus v n → 0 in L qs ′ q (∂Ω).By uniqueness of the limit we have .
Passing to the limit, we obtain a contradiction.✷ Proof of Lemma 3.4.Let C r the constant from embedding W 1,r (Ω) ⊂ L rs ′ r (∂Ω), where r = p, q.According to Lemma 3.5, there exists C(d) > 0 such that where d such that For any u ∈ X(d) satisfying ∂Ω m q u q + dσ ≤ 0 by (3.16) and (3.15) we have For any u / ∈ X(d) satisfying ∂Ω m q u q + dσ ≤ 0 thanks to (3.13) and (3.16) we find On a Positive Solution for (p, q)-Laplace Equation . . .
Proof.Taking into account that λ > λ 1 (p, m p ) and p > q, we claim that (3.23) is true because for a sufficiently large R > 0 we have  This completes the proof of Theorem 3.1.
Then the functional I λ,mp,mq satisfies the Palais-Smale condition.

✷
Recalling that I λ,mp,mq satisfies the Palais-Smale condition by virtue of Lemma 3.3, the properties pointed out in (3.12) and (3.23) allow us to apply the mountain pass theorem, which guarantees the existence of a critical value c ≥ δ of I λ , with δ > 0 in (3.12), namely c := inf γ∈Σ max t∈[0,1]