Existence of Solutions for a Class of p-Laplacian Type Equation with Critical Growth and Potential Vanishing at Infinity

In this paper, we study the existence of positive 
solution for the following p-Laplacain type equations with critical nonlinearity 
 \begin{equation*} 
 \left\{ 
 \renewcommand{\arraystretch}{1.25} 
 \begin{array}{ll} 
 -\Delta_p u + V （x)|u|^{p-2}u = K(x)f(u)+P(x)|u|^{p^*-2}u, \quad 
 x\in\mathbb{R}^N,\\ 
 u \in \mathcal{D}^{1,p}(\mathbb{R}^N), 
 \end{array} 
 \right. 
 \end{equation*} 
where $\Delta_p u = div(|\nabla u|^{p-2} \nabla u),\ 1 < p < N,\ p^* =\frac 
{Np}{N-p}$, $V(x)$, $K(x)$ are positive continuous functions which vanish at 
infinity, $f$ is a function with a subcritical growth, and $P(x)$ is a bounded, 
nonnegative continuous function. 
By working in the weighted Sobolev spaces, and using variational method, we 
prove that the given problem has at least one positive solution.

1. Introduction and main results. In this paper, we study the existence of positive solution for the following p-Laplacain equation with critical nonlinearity −∆ p u + V (x)|u| p−2 u = K(x)f (u) + P (x)|u| p * −2 u, x ∈ R N , u ∈ D 1,p (R N ), (1.1) where 1 < p < N , p * = N p N −p is the critical Sobolev exponent, the potential V (x) and K(x) : R N → R are positive continuous functions vanishing at infinity, f : R → R is a function with a subcritical growth and P (x) ≥ 0 is a bounded continuous function.
When p = 2, problem (1.1) appears in many interesting physical contexts. For example, the solutions of this class of problems are related to the existence of standing wave solutions ψ(x, t) = exp(−iE 0 t)v(x) for nonlinear Schrödinger equation where E 0 ∈ R and v(x) is a real function. Obviously, ψ satisfies (1.2) if and only if the function v(x) solves the semilinear scalar field equation here we set f (u) = |u| −1 g(|u|)u. Problem of (1.3) has been extensively studied since 1970s, we would like to mention [6,7,16] to the readers. An important class of problems associated with (1.3) is the zero mass case, which occurs with the potentials V (x) vanishing at infinity, that is, In [2], Ambrosetti, Felli and Malchiodi studied (1.3) with the zero mass case when f (s) = s p with 2 < p < N + 2 N − 2 and V , K satisfying the following assumptions: V , K: R N → R are smooth functions and there exist constants α, β, a, A, κ > 0 such that a 1 + |x| α ≤ V (x) ≤ A, and 0 < K(x) ≤ and α, β verifying < p, if 0 < β < α, or p > 1, when β ≥ α.
The condition (V K) is interesting because Opic and Kufner in [19] have showed that it can be used to prove that the space E given by E = u ∈ D 1,2 (R N ) : R N V (x)u 2 dx < +∞ endowed with the norm is compactly embedded into the weighted Lebesgue space L p+1 K (R N ) = u : R N → R u is measurable and R N K(x)|u| p+1 dx < +∞ .
Ambrosetti and Wang [4] also considered the condition (V K) but the inequality on V (x) is assumed only outside of a ball centered at origin. Alves and Souto [1] considered more general conditions on V (x) and K(x), by which the space E can compactly embedded into the weighted space L p+1 K (R N ) for certain p ∈ [1, 2 * − 1). Moreover, Bonheure and Schaftingen [8] introduced a new set of hypotheses on V (x) and K(x), by using the Marcinkiewicz spaces L r,∞ (R N ) for r > 1, which permitted them to show continuous and compact embeddings from E into the weighted space L q K (R N ) for some q > 1.
The space E with fast increasing potentials also studied by many authors. For example, set K(x) = exp(|x| 2 /4), the space is used as the working space to deal with the following elliptic problem For the reference, we would like to mention [9,14,15] and the reference therein.
In particular, Escobedo and Kavian [14] have proved that the embedding E → L q K (R N ) is continuous for all q ∈ [2, 2 * ] and it is compact for all q ∈ [2, 2 * ). Over the last several decades, many authors have shown much interest in the second-order elliptic differential equations in unbounded domains with critical growth. For example, P. L. Lions [17,18] established a Concentration-Compactness Principle for some nonlinear elliptic equations in R N and studied minimization problems associated with nonlinear elliptic equations in R N with critical growth. D. Smets [21] investigated the following problems satisfying some conditions. He established a complete noncompact analysis of (1.4) and through this analysis he obtained some existence results of the solutions for (1.4). In the non-compact analysis, he divided any Palais-Smale sequence {u n } of the variational functional corresponding to (1.4) into two parts. One part is confined in a ball B(R) and the other part is confined in R N \B(R). As for the part confined in a ball, he can treat it as the case that {u n } is tight. For the other part, it can be transformed into one which confined in a ball by Kelvin transformation. The Kelvin transformation can be applied to problem (1.4) because there are no new blow up bubbles, except for the blow up bubbles caused by critical nonlinearity and Hardy term, to occur for the Palais-Smale sequences of the corresponding variational functional of (1.4).
Recently, Deng, Jin and Peng [11] established a complete non-compact expression for the Palais-Smale sequences of the variational functional corresponding to (1.5) which including all the blow up bubbles caused by critical exponents, Hardy term and unbounded domains. By using the non-compact expression for the Palais-Smale sequences of the variational functional corresponding to (1.5), the existence of positive solutions for problem (1.5) is obtained. But the potential a(x) should be non-vanishing at infinity. For more results about the existence of solutions for semilinear or qusilinear elliptic problems with critical growth and non-vanishing potential at infinity, the readers can refer to the papers [10,12,13,16,23,26] and the reference therein. However, there seems to be little progress on the existence of positive solution for a general elliptic equation, for example problem (1.1), with critical growth and the potential V (x) vanishing at infinity.
In this paper, we establish the existence of positive solution for problem (1.1) with critical nonlinearity and the potential V (x) vanishing at infinity. To this end, we need some assumptions on V (x), K(x), f (s) and P (x).
As in [1], we say (V, K) ∈ K if the following conditions hold: (ii) If {A n } ⊂ R N is a sequence of Borel sets such that |A n | ≤ R, for all n and some R > 0, we have lim r→+∞ An∩B c r (0) K(x)dx = 0, uniformly in n ∈ N.
(iii) One of the following conditions occurs: or there is p 0 ∈ (p, p * ) such that Related to the function f , we assume the following conditions: Moreover, for the function P (x), we assume that (P 1 ) There is a point x 0 , such that Our main result of this paper is as follows: There are serious difficulties in trying to find the nontrivial solutions in D 1,p (R N ) of (1.1) by standard variational methods. Firstly, the potential V (x) vanishes at infinity, we cannot work on the usual Sobolev space W 1,p (R N ). Secondly, the space D 1,p (R N ) can not embedding into L r (R N ) for p ≤ r < p * and the embedding D 1,p (R N ) → L p * (R N ) is not compact. In order to prove the existence result, we first define space E, which is a subspace of D 1,p (R N ), and weight space L q K (R N ). We then establish a Hardy-type inequality (see Lemma 2.1) involving V (x) and K(x) as in [1]. Since the embedding E → L p * P (R N ) is still not compact, we imitate the method in [7] by using the mountain pass theorem without (P S) condition, and the existence of positive solution in E (also in D 1,p (R N )) of (1.1) is proved. The rest of this paper is organized as follows. In Section 2, we present some embedding results, which generalize the corresponding embedding results in [1]. In Section 3, we prove our main result.
2. Some preliminary lemmas. In order to prove the main results, first we introduce the space E and L q K (R N ) are particular cases of weighted space and are discussed in [19]. The following two lemmas discuss the continuous and compact embedding for E → L q K (R N ). The proof is inspired by [1,8].
Proof. By assuming that (K 2 ) is true, the proof is trivial if q = p or p * . Now, we fix q ∈ (p, p * ) and let λ = p * −q p * −p , then q = λp + (1 − λ)p * . It follows that
Proof. The proof of this lemma will be divided into two parts.
Firstly, we consider the case when (K 2 ) holds. For fixed q ∈ (p, p * ) and given ε > 0, there are 0 < s 0 < s 1 and C > 0 such that On the other hand, setting the last inequality implies that which shows that sup n∈N |A n | < +∞. Therefore, from (K 1 ), there is an r > 0 such that An∩B c r (0) for all n ∈ N. Once that q ∈ (p, p * ) and K is a continuous function, it follows from Sobolev embeddings on the bounded domain that lim n→+∞ Br(0) Combining (2.4) and (2.5), , ∀q ∈ (p, p * ). Now, we suppose that (K 3 ) holds. First of all, it is important to observe that for each x ∈ R N fixed, the function Hence, ∀x ∈ R N and s > 0. Combining the last inequality with (K 3 ), there is r > 0 large enough, such that K(x)|s| p0 ≤ Cε(V (x)|s| p + |s| p * ), ∀s ∈ R and |x| ≥ r, which leads to and so Since p 0 ∈ (p, p * ) and K is a continuous function, it follows from Sobolev embedding on bounded domain that lim n→+∞ Br(0) which implies that v n → v in L p0 K (R N ). Then the proof of our lemma is completed.
Proof. We only give the proof of (2.10) and the proof of (2.11) can be proved in the same way. We begin the proof by assuming that (K 2 ) occurs. From (f 1 )-(f 2 ), fixed q ∈ (p, p * ) and given ε > 0, there is C > 0 such that and there is r > 0 such that Combining the last inequalities with (2.12) and (2.13), (2.14) Now we assume (K 3 ) holds. Repeating the same arguments explored in the proof of lemma 2.2, for given ε > 0 small enough, there is r > 0 large enough such that K(x) ≤ ε(V (x)|s| p−p0 + |s| p * −p0 ), ∀s ∈ R and |x| > r.
From (f 1 ) and (f 2 ), for the given ε > 0, we have F (s) ≤ C|s| p0 + ε|s| p * , ∀s ∈ I, , then for all s ∈ I and |x| > r, we have Therefore, for any u ∈ E, we have the following estimate Thus,

Repeating the same arguments used in the proof of lemma 2.2, it follows that
An∩B c r (0) K(x)dx → 0 as r → +∞, and so, for n large enough To conclude the proof, we need to show that lim n→+∞ Br(0) However, this limit follows by using the compactness lemma of Strauss [see [22], Compactness lemma 2, p.156]: B r (0) is a bounded domain, |v n | L p * (Br(0)) is bounded and (f 2 ), together with the convergence almost everywhere imply the limits as required.
3. The proofs of our main results. In this section, we prove the existence of positive solutions for problem (1.1) by variation technique. In order to use mountain pass lemma [7] to obtain the solution, some arguments of this proof were adapted from the articles [7,12,13].
Since we intend to find positive solutions, we assume that The variational functional associated with (1.1) is given by for all u ∈ E. From the conditions on f (s) and Lemma 2.1, the functional I is well defined and I ∈ C 1 (E, R). Its Gateaux derivative is given by Then, it is easy to check that the critical points of I are weak solutions of (1.1). Since E can be embedded into L q K (R N ) continuously for some q (see Lemma 2.1), we can verify that the functional I exhibits the Mountain-Pass geometry. (ii) there exists an e ∈ E, such that u > ρ and I(e) < 0.
As a consequence of Lemma 3.1 and the mountain pass lemma (see [3]), for the constant there exists a (P S) c sequence {u n } in E at the level c 0 , that is, Proof. From (f 3 ) we have Since I(u n ) → c 0 and I (u n ) → 0 as n → +∞, we deduce that {u n } is bounded in E.
Let {u n } be a (P S) c0 -sequence given by (3.4). By Lemma 3.2, we deduce that {u n } is bounded. Using the standard argument (see [5,12,16,25,26]), up to a subsequence, there is u ∈ E such that (3.5) In the following, we show that u is a positive solution of (1.1). To this end, we exploit the fact that the critical equation We can choose k > 0 such that where S denote the best constant for the embedding D 1,p (R N ) → L p * (R N ), namely Let R > 0 be small enough that B 2R (x 0 ) ⊂ R N , and ψ(x) be a piecewise smooth function with support in and Since ∂u /∂ n ≤ 0, we have that and by the assumption (P 2 ) we also have Simple calculations as [12] gives that as ε → 0, where k is a positive constant. Therefore, we have |∇v ε | p dx, since for small ε > 0, say ε ≤ ε 0 , it is easy to see that for some positive constant C ε0 . The definition of V ε and the last two inequalities imply that Lemma 3.3. Assume that (V, K) ∈ K, f satisfies (f 1 )-(f 3 ) and P (x) satisfies (P 1 )-(P 2 ), Then there exists a u 0 ∈ E\{0} such that if either N ≥ p 2 or p < N < p 2 and θ > p * − p p−1 . Proof. We consider now Clearly, lim t→+∞ I(tv ε ) = −∞ for all ε > 0, by (f 1 ) and (f 2 ), sup t≥0 I(tv ε ) > 0 is attained by some t ε > 0. We claim that there are two positive constants A 1 and A 2 independent of ε such that A 1 < t ε < A 2 if ε > 0 is sufficiently small.
Next, we suppose that (K 3 ) holds. By (f 1 ), (f 2 ), there is a constantC > 0, such that It follows from (3.11) that p for all t ≥ 0, the estimate (3.9) on V ε and the above inequality imply that (3.12) By (f 3 ), we have that F (s) ≥ Cs θ for s > 0. Therefore It follows from (3.12) and the definitions of v ε and ψ that (3.13) If N ≥ p 2 and θ ∈ (p, p * ), we have It follows from (3.13) that if we choose ε small enough. If p < N < p 2 and θ ∈ (p * − p p−1 , p * ), we have Then inequality (3.14) also follows from (3.13) if we choose ε small enough. Now we can imply that the inequality (3.10) holds by taking u 0 = v ε for sufficiently small ε.
The proof of Theorem 1.1. The conditions for the Mountain Pass Lemma [7] are satisfied by Lemma 3.1. From (3.2)-(3.4), we have and Denote v n = u n − u, then from (3.5), Lemma 2.3 and Brezis-Lieb Lemma [5], and Since I (u n ) → 0 as n → ∞, and by (3.5) again, we have  and Without loss of generality we can suppose R N (|∇v n | p + V (x)|v n | p )dx → as n → ∞, (3.22) and from (3.20) we have that R N P (x)|v n | p * dx → as n → ∞. has at least one positive solution u such that 0 < I(u) < 1 N S N p if either N ≥ p 2 or p < N < p 2 and θ > p * − p p−1 .