A variational approach to a quasilinear multiparameter elliptic system involving the p-Laplacian and nonlinear boundary condition

We present a note on the paper by Brown and Wu (J Math Anal Appl 337:1326–1336, 2008). Indeed, we extend the multiplicity results for a class of semilinear elliptic system to the quasilinear elliptic system of the form: $$\left\{\begin{array}{ll}-\Delta_p u + m(x)\,|u|^{p-2}u = \frac{\alpha}{\alpha + \beta} \, |u|^{\alpha -2} \,u\,|v|^{\beta}, \quad\quad\quad\quad\quad\quad\quad \,\,\, x\in \Omega,\\ -\Delta_p v + m(x)\,|v|^{p-2}v = \frac{\beta}{\alpha + \beta} \, |u|^{\alpha} \,|v|^{\beta -2} \,v, \quad\quad\quad\quad\quad\quad\quad \,\,\, x\in \Omega,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial n} = \lambda\,a(x)|u|^{\gamma-2}u, \,\,\,|\nabla v|^{p-2} \frac{\partial v}{\partial n} = \mu\,b(x)|v|^{\gamma-2}v, \quad x\in \partial \Omega. \end{array} \right.$$Here Δp denotes the p-Laplacian operator defined by $${\Delta_{p}z=div\,(|\nabla z|^{p-2} \nabla z), p >2 , \Omega \subset \mathbb{R}^N}$$ is a bounded domain with smooth boundary, $${\alpha >1 , \beta >1 , 2 < \alpha+\beta < p < \gamma < p* (p* = \frac{pN}{N-p} {\rm if} N > p, p*=\infty {\rm if} N \,\leq p), \frac{\partial}{\partial n}}$$ is the outer normal derivative, $${(\lambda, \mu) \in \mathbb{R}^{2} {\setminus} \{(0,0)\},}$$ the weight m(x) is a positive bounded function, and $${a(x), b(x)\in C(\partial \Omega)}$$ are functions which change sign in $${\overline{\Omega}.}$$

Problems involving the s-Laplace operator arise in some physical models like the flow of non-Newtonian fluids: pseudo-plastic fluids correspond to s ∈ (1, 2) while dilatant fluids correspond to s > 2. The case s = 2 expresses Newtonian fluids [6]. On the other hand, quasilinear elliptic systems like (1) has an extensive practical background. It can be used to describe the multiplicate chemical reaction catalyzed by the catalyst grains under constant or variant temperature, it can be used in the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary (see [18,22]) and can be a simple model of tubular chemical reaction, more naturally, it can be a correspondence of the stable station of dynamical system determined by the reaction-diffusion system, see Ladde and Lakshmikantham et al. [19]. More naturally, it can be the populations of two competing species [16]. So, the study of positive solutions of elliptic systems has more practical meanings. We refer to [8,15] for additional results on elliptic systems.
In recent years, several authors have used the Nehari manifold and fibering maps (i.e., maps of the form t −→ J λ (tu) where J λ is the Euler function associated with the equation) to solve semilinear and quasilinear problems (see [1][2][3]9,[11][12][13][14]17,[23][24][25][26]). By the fibering method, Drabek and Pohozaev [17], Bozhkov and Mitidieri [9] studied, respectively, the existence of multiple solution to a p-Laplacian single equation and ( p, q)-Laplacian system. Brown and Zhang [14] have studied the following subcritical semilinear elliptic equation with a sign-changing weight function where γ > 2. Exploiting the relationship between the Nehari manifold and fibering maps, they gave an interesting explanation of the well-known bifurcation result. In fact, the nature of the Nehari manifold changes as the parameter λ crosses the bifurcation value. Recently, in [11], the author considered the above problem with 1 < γ < 2. In this work, we give a variational method which is similar to the fibering method (see [17,14] or [14]) to prove the existence of at least two nontrivial nonnegative solutions of Problem (1). In particular, by using the method of [13], we do this without the extraction of the Palais-Smale sequences in the Nehari manifold as in [1,3]. This paper is divided into three sections, organized as follows. In Sect. 2, we give some notation, preliminaries, properties of the Nehari manifold and set up the variational framework of the problem. In Sect. 3, we give our main results.

Variational setting
which is equivalent to be the standard one. Throughout this paper, we set C 1 and C 2 be the best Sobolev and the best Sobolev trace constants for the embedding of W 1, p 0 ( ) in L γ (∂ ) and W 1, p 0 ( ) in L α+β ( ), respectively. First, we give the definition of the weak solution of (1).

Definition 2.1
We say that (u, v) ∈ W is a weak solution to (1) if for all (w 1 , w 2 ) ∈ W we have It is clear that Problem (1) has a variational structure. Let I λ,μ : W → R be the corresponding energy functional of Problem (1) is defined by It is well known that the weak solutions of Eq. (1) are the critical points of the energy functional I λ,μ . Let J be the energy functional associated with an elliptic problem on a Banach space X. If J is bounded below and J has a minimizer on X, then this minimizer is a critical point of J. So, it is a solution of the corresponding elliptic problem. However, the energy functional I λ,μ , is not bounded below on the whole space W, but is bounded on an appropriate subset, and a minimizer on this set (if it exists) gives rise to solution to (1).
Then we introduce the following notation: for any functional f : Consider the Nehari minimization problem for (λ, μ) ∈ R 2 \{(0, 0)}, It is clear that all critical points of I λ,μ must lie on S λ,μ which is known as the Nehari manifold (see [16]). We will see below that local minimizers of I λ,μ on S λ,μ are usually critical points of I λ,μ . It is easy to see that Note that S λ,μ contains every nonzero solution of Problem (1). Define Then for (u, v) ∈ S λ,μ , Now, we split S λ,μ into three parts: To state our main result, we now present some important properties of S + λ,μ , S 0 λ,μ , and S − λ,μ .

Lemma 2.2
There exists ζ 0 > 0 such that for By the Sobolev trace imbedding theorem, and So, there exists 0 such that Then using the Hölder inequality and the Sobolev inequality, we get which is a contradiction. Thus, we can conclude that there exists ζ 0 > 0 such that for 0 < (|λ| a ∞ ) Lemma 2. 3 We have Proof (i) We consider the following two cases: (ii) We consider the following two cases: It follows that the conclusion is true.
As proved in Binding et al. [7] or in Brown and Zhang [14], we have the following lemma.
Then we have the following result.

Lemma 2.5 I λ,μ is coercive and bounded below on S λ,μ .
Proof If (u, v) ∈ S λ,μ , it follows from (2) and the Sobolev embedding theorem Thus I λ,μ is coercive and bounded below on S λ,μ .
Then we have the following lemma.

Lemma 2.7 For each
. Then E(t) achieves its maximum at t max , increasing for t ∈ [0, t max ) and decreasing for t ∈ (t max , ∞). Moreover, (12) and We This completes the proof.
For each u ∈ W with R(u, v) > 0, we write Then we have the following lemma. N (u, v) ≤ 0, then there is a unique t + < t max such that (t + u, t + v) ∈ S + λ,μ and

Lemma 2.8 For each u ∈ W with R(u, v) > 0, we have
Clearly, . Then E(t) achieves its maximum at t max , increasing for t ∈ [0, t max ) and decreasing for t ∈ (t max , ∞). Using the argument in Lemma (2.7) we can obtain the result of Lemma 2.8

Existence of solutions
Now we can state our main results.
The proof of Theorem (3.2) is similar to that of Theorem (3.1) and for this reason, will be omitted here.

Remark 3.3
Our ideas can also be applied to the following elliptic system: where p, α, β, γ, m(x), a(x) and b(x) are as before. The results presented here have analogous statements for the latter problem. The proofs of the multiplicity results are similar to the ones performed for Problem (1) so we leave the details to the reader.
The proof of the Theorem (3.1) will be a consequence of the next two propositions.

This implies
and by Theorem 2.6 (i) Letting n → ∞, we see that R(u 0 , v 0 ) > 0. In particular u + 0 = 0, v + 0 = 0. Now we prove that u n → u + 0 strongly in W where E(t) is as in (14). Clearly, K (u,v) (t) → −∞ as t → 0 + , and , by an argument similar to the one in the proof of Lemma (2.8) we have that the function K (u,v) (t) achieves its maximum at t max , is increasing for t ∈ (0, t max ) and decreasing for t ∈ (t max , ∞), where is as in (13).
Next, we establish the existence of a local minimum for I λ,μ on S − λ,μ .
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