Solutions for the quasi-linear elliptic problems involving the critical Sobolev exponent

In this article, we study the existence and multiplicity of positive solutions for the quasi-linear elliptic problems involving critical Sobolev exponent and a Hardy term. The main tools adopted in our proofs are the concentration compactness principle and Nehari manifold.

In this paper, we use the following norm of W  p , the usual norm in W ,p  ( ). The norm in L p ( ) is represented by u p = ( |u| p dx)  p . According to Hardy inequality, the following best Sobolev constant is well defined for  < p < N , and  ≤ μ < μ: The quasi-linear problems on Hardy inequality have been studied extensively, either in the smooth bounded domain or in the whole space R N . More and more excellent results have been obtained, which provide us opportunities to understand the singular problems. However, compared with the semilinear case, the quasi-linear problems related to Hardy inequality are more complicated [-]. Abdellaoui, Felli and Peral [] considered the extremal function which achieves the best constant S μ, , and gave the properties of the extremal functions. The conclusions obtained in [] can be applied in the problems with critical Sobolev exponent and Hardy term.
Wang, Wei and Kang [] investigated the following problem: where  < q < p, μ > , f and g are non-negative functions and p * = Np N-p is the critical Sobolev exponent. The property of the Nehari manifold was used to prove the existence of multiple positive solutions for (.). Furthermore, Hsu [, ] improved and complemented the main results obtained in []. Recently, Goyal and Sreenadh [] investigated a class of singular N -Laplacian problems with exponential nonlinearities in R N . Very recently, Xiang [] established the asymptotic estimates of weak solutions for p-Laplacian equation with Hardy term and critical Sobolev exponent.
We should mention that Liu, Guo and Lei [] studied the existence and multiplicity of positive solutions of Kirchhoff equation with critical exponential nonlinearity. Inspired by [, ], we study the problem (.) on critical Sobolev exponent. Comparing with the main results obtained in [, , -], in this paper, on the one hand, we will analysis the effect of β|x| α-p |u| p- u, and the more careful estimates are needed. On the other hand, we establish an lower bound for λ * (λ * is defined in Theorem .).
Define the energy functional associated to problem (.) as follows: We obtain the following result.
Theorem . Suppose that  < q < p,  < α < p -. Then there exists λ * >  such that problem (.) admits at least two solutions and one of the solutions is a ground state solution for all λ ∈ (, λ * ).

Preliminaries
Firstly, we introduce the Nehari manifold Furthermore u ∈ N λ if and only if N λ can be divided into the following three parts: Applying the Hölder inequality and the Sobolev inequality, for all u ∈ W ,p  ( )\{} we have Let (s) = , that is, We can deduce that It is easy to check that (s) >  for all  < s < s max and (s) <  for all s > s max . Consequently, (s) attains its maximum at s max , that is, By (.) and (.), we have where  < λ < T  . Thus, there exist constants s + and ssuch that (ii) We prove that N  λ = ∅ for all λ ∈ (, T  ). By contradiction, assume that there exists that is, Note that (.) holds for u ∈ N  λ \{}. Then It follows from (.) and (.) that for  < λ < T  . This is a contradiction.

Lemma . I λ is coercive and bounded below on N λ .
Proof For u ∈ N λ , we can deduce from (.) and (.) that Note that  < q < p and  < β < β  , we see that I λ is coercive and bounded below on N λ .
as follows: It is clear that which implies that Lemma . tells us that F s (, ) = . Thus, by the implicit function theorem at the point (, ), there exist ε > , and a differentiable function Proof The proof is similar to that of Lemma ., and we omit it here.
Proof It follows from Lemma . that I λ is coercive on N λ . Using the Ekeland variational principle [], we can find a minimizing sequence {u n } ⊂ N λ of I λ satisfying Without loss of generality, we can assume that u n ≥ . By Lemma ., we know that {u n } is bounded in W ,p  ( ). As a consequence, there exist a subsequence (still denoted by {u n }) and u * in W ,p  ( ) such that (.) From Lemma ., for s >  sufficiently small and φ ∈ W ,p  ( ), and set u = u n , ω = sφ ∈ W ,p  ( ), we can find that f n (s) = f n (sφ) such that f n () =  and f n (s)(u n + sφ) ∈ N λ . Since Notice that Therefore Dividing by s >  and taking the limit for s → , combining with (.) and (.), we have for every φ ∈ W ,p  ( ). Note that (.) holds equally for -φ, we see that (.) holds.
Then the limiting problem has radially symmetric ground states where the function U p,μ (x) = U p,μ (|x|) is the unique radial solution of the above limiting problem with In the following, we define =  N S N p μ, .
(  .   ) Then there exists u ∈ W ,p  ( ) such that u n → u in L p * ( ).

Proof
Since By Lemma ., we know that {u n } is bounded in W ,p  ( ). In fact, we can deduce from (.) and (.) that where  < β < β  ,  < q < p, we see that {u n } is bounded in W In term of the concentration compactness principle, going if necessary to a subsequence, there exist an at most countable set J , a set of points {x j } j∈J ⊂ \ {}, and real numbers μ j , ν j , χ  such that where δ x j is the Dirac mass at x j . Let be sufficient small satisfying  / ∈ B(x j , ) and B(x j , ) ∩ B(x i , ) = ∅ for i = j, i, j = , , . . . , k. Let ψ ,j (x) be a smooth cut-off function centered at x j such that  ≤ ψ ,j (x) ≤ , ψ ,j (x) =  for x ∈ B(x j ,  ), ψ ,j (x) =  for x ∈ \B(x j , ) and |∇ψ ,j (x)| ≤  . Note that Furthermore, we have By (.), we deduce that where D is defined in (.). Hence, we conclude that -Dλ p p-q ≤ κλ < -Dλ p p-q , which is a contradiction. It follows that ν j =  for j ∈ {} ∪ J , which means that |u n | p * dx → |u| p * dx as n → ∞. The proof is completed.