Existence of solution for a quasilinear elliptic Neumann problem involving multiple critical exponents

In this paper, we study the Neumann boundary value problem to a quasilinear elliptic equation with the critical Sobolev exponent and critical Hardy–Sobolev exponent, and prove the existence of nontrivial nonnegative solution by means of variational method.


Introduction and main results
In this paper, we discuss the following quasilinear elliptic problem with critical Sobolev exponent and critical Hardy-Sobolev exponent: ⎧ ⎨ ⎩ p u + λ|u| p-2 u = Q(x)|u| p * -2 u + P(x) |u| p * (t)-2 u |x| t , x ∈ Ω, |∇u| p-2 ∂u ∂ν = 0, x ∈ ∂Ω\{0}, (1.1) where Ω ⊂ R N is a bounded domain with ∂Ω ∈ C 2 , 0 ∈ ∂Ω, p u = div(|∇u| p-2 ∇u) is the p-Laplace operator, 1 < p < N , p * = Np N-p , p * (t) = p(N-t) N-p (0 ≤ t < p) is the so-called critical Hardy-Sobolev exponent, and ν denotes the unit outward normal vector with respect to ∂Ω, λ ∈ R is a parameter, the weight functions Q(x), P(x) are continuous on Ω. Such problems arise in the theory of quasiregular and quasiconformal mapping or in the study of non-Newtonian fluids. In the latter case, the parameter p is a characteristic of the medium.
Media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics.
If p = 2, they are Newtonian fluids.
The study of semilinear elliptic problems with critical growth terms is one of hot spots in partial differential equations. In the case of p = 2, problem (1.1) is transformed into the following semilinear elliptic problem: (1.2) When functions Q(x) = 1, P(x) = 0, Comte and Knaap [1] established the existence of nontrivial solution for problem (1.2) by variational method, while Adimurthi et al. [2] proved the existence of a nonradial positive solution. Chabrowski and Willem [3] obtained the existence of least energy solutions by solving minimization problem corresponding to is nonnegative and Hölder continuous. Subsequently, Chabrowski [4] proved the existence of at least two solutions to (1.2) by the mountain pass principle if one of the weight functions changes sign. In the paper [5], the authors studied the following semilinear elliptic problem with Hardy-Sobolev exponent: they obtained the existence of positive solutions by the mountain pass lemma without (PS)-condition and the strong maximum principle. Other related results on the semilinear elliptic problems can be seen in [6][7][8][9][10][11][12] and the references therein. As for the quasilinear elliptic problems with critical Sobolev or Hardy-Sobolev exponents, the existence and multiplicity of solutions have also been studied extensively. Abreu et al. [13] studied the following the nonhomogeneous Neumann boundary problem: where p -1 < q ≤ p * -1, ϕ ∈ C α (Ω), 0 < α < 1, ϕ ≡ 0. They proved that there exists a λ * > 0 such that problem (1.4) has at least two positive solutions if λ > λ * , has at least one positive solution if λ = λ * , and has no positive solution if λ < λ * relying on the lower and upper solutions method and variational approach. Subsequently, Deng and Jin [14] also obtained the existence of solutions to problem (1.4) if u q is replaced by u q |x| t , where p -1 < q ≤ p * (t) -1, 0 ≤ t < p -1. Li and Xia [15] studied the existence of multiple solutions for quasilinear Neumann problem with critical Sobolev exponent. With regard to the multiple critical exponents, Filippucci et al. [16] investigated the quasilinear elliptic problem involving multiple critical terms on the whole space and obtained the existence of positive solutions by using the existence of extremals of some Hardy-Sobolev type embedding. Li et al. [17] showed the existence and multiplicity of solutions to the quasilinear elliptic equations with Dirichlet boundary conditions and combined critical Hardy-Sobolev terms on bounded smooth domains by employing Ekeland's variational principle.
However, as far as we know, there are few results of the Neumann boundary condition for quasilinear elliptic equations with the critical Sobolev exponent and critical Hardy-Sobolev exponent. Motivated by the results of the above-mentioned papers, in this paper we aim to show the existence of nontrivial nonnegative solution to problem (1.1) by the variational method. The special features of this problem are the following. Firstly, due to the lack of compactness of the embedding of W 1,p (Ω) → L p * (Ω) and W 1,p (Ω) → L p * (t) (Ω, |x| -t ), we cannot use the standard variational argument directly. In order to overcome this difficulty and obtain the existence of solutions, we have to add restrictions on the weight functions Q(x) and P(x) to prove that the corresponding functional of problem (1.1) satisfies the (PS) c -condition in a suitable range by the Lions concentration-compactness principle. Secondly, when λ ≤ 0, the weight function P(x) is allowed to change sign and the space W 1,p (Ω) is not suitable for our problem. In order to obtain the existence of solution, we have to introduce a suitable space. This result is an extension of a work by Li and Xia [15].
Throughout this paper, we define , the main results of this paper are the following theorems.   |x| t dx < 0. Then there exists a constant μ * > 0 such that Problem (1.1) has at least one nontrivial nonnegative solution for all 0 ≤ μ < μ * , where μ = -λ, and μ * will be given in Sect. 4.
The outline of this paper is as follows. In Sect. 2, we present some necessary preliminaries. Section 3 is devoted to the proofs of Theorems 1.1 and 1.2. The proof of Theorem 1.3 is given in Sect. 4.

Preliminaries
Let W 1,p (Ω) be the Sobolev space with norm u = ( Ω (|∇u| p +|u| p ) dx) 1 p , the best Sobolev constant and the best Hardy-Sobolev constant are defined by The constant S is achieved by the functional u ε given by Next, we give the definition of the weak solution to Problem (1.1).
The corresponding nonnegative solutions of Problem (1.1) are equivalent to the critical points of the energy functional In order to obtain the existence of solution to Problem (1.1), we need the following lemmas.

Lemma 2.2 ([14]
(Hardy-Sobolev inequality)) Assume that 1 < p < N and p * (t) = p(N-t) N-p , 0 ≤ t ≤ p. Then there exists a constant C > 0 such that, for any u ∈ W 1,p (Ω), By Sobolev's inequality, there exists a constant C 1 > 0 such that Now, we prove that the energy functional J λ (u) satisfies the geometry of mountain pass lemma.
We define Using Lemma 2.3 and the mountain pass lemma, there exists a (PS) c -sequence {u n } ⊂ W 1,p (Ω) such that J λ (u n ) → c, J λ (u n ) → 0 as n → ∞. Then we have the following lemma.
Combining (2.5) with (2.6), it follows from the property of function Q(x) that thus, we can get that {u n } is bounded in W 1,p (Ω). Next, we prove that {u n } is relatively compact in W 1,p (Ω). Since {u n } is bounded in W 1,p (Ω), we know that there exists a subsequence, still denoted by {u n }, such that u n u weakly in W 1,p (Ω), u n u weakly in L p * (Ω), u n u weakly in L p * (t) Ω, |x| -t , u n → u strongly in L p (Ω), u n → u almost everywhere in Ω.
By the Lions concentration-compactness principle [27], there exist at most countable set J, a set of distinct points {x j } j∈J ⊂ Ω\{0}, sets of nonnegative real numbers {μ j } j∈J , {ν j } j∈J and nonnegative real numbers μ 0 , ν 0 , γ 0 such that in the weak sense of measure, where δ x is the Dirac mass at x, and the constants μ j , ν j , γ 0 satisfying Now, we prove μ j = 0 and ν j = 0, where j ∈ J. In fact, we choose ε > 0 sufficiently small Noting that and by (2.7), we have that Thus, If ν j = 0, by (2.8) and (2.9), we find that On the other hand, , which is a contradiction. Hence, μ j = ν j = 0.

Proofs of Theorems 1.1 and 1.2
In order to prove Theorems 1.1 and 1.2, we also need the following lemma.

Lemma 3.1 Assume that λ > 0 and the coefficients Q(x), P(x) are positive continuous functions on Ω, then there exists
Proof For u ∈ W 1,p (Ω) \ {0}, we consider the functional Proof of Theorem 1.1 Applying Lemma 2.4, we know that the functional J λ (u) satisfies the (PS) c -condition, by Lemma 3.1 and the mountain pass theorem, we obtain that Problem (1.1) has at least one nontrivial solution u. On the other hand, since J λ (u) = J λ (|u|), then Problem (1.1) has at least one nontrivial nonnegative solution. The proof of Theorem 1.1 is completed.
In order to prove Theorem 1.2, we also need the following lemma.