Positive ground state solutions for a quasilinear elliptic equation with critical exponent

In this paper, we study the following quasilinear elliptic equation with critical Sobolev exponent: \begin{document} $ -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=|u|^{2^*-2}u+|u|^{p-2}u, \quad x\in {{\mathbb{R}}^{N}}, $ \end{document} which models the self-channeling of a high-power ultra short laser in matter, where N ≥ 3; 2 14 ], we obtain the existence of positive ground state solutions for the given problem.

1. Introduction and main results. In this paper, we study the existence of positive ground state solutions for the following quasilinear elliptic equation: = |u| 2 * −2 u + |u| p−2 u, x ∈ R N , (1) where N ≥ 3, 2 < p < 2 * with 2 * = 2N N −2 the critical Sobolev exponent, and V (x) is a given positive potential.
The solutions of this type of quasilinear elliptic equation are related to the solitary wave solutions for quasilinear Schrödinger equations of the form i∂ t z = −∆z + W (x)z − h(z) − ∆l(|z| 2 )l (|z| 2 )z, (2) where z : R × R N → C, W : R N → R is a given potential and h, l : R → R are suitable functions. Quasilinear equations (2) have been derived as models of several physical phenomena and have been the subject of extensive study in recent years. The case l(s) = s models the time evolution of the condensate wave function in super-fluid film ( [16,17]). This equation has been called the superfluid film equation in fluid mechanics by Kurihara [16]. In the case l(s) = (1+s) 1/2 , problem (2) models the self-channeling of a high-power ultrashort laser in matter, see [4,6,8,27]. Set z(t, x) = exp(−iEt)u(x), where E ∈ R and u is a real function, (2) can be reduced to the corresponding equations of elliptic type: If we set l(s) = s, we get the superfluid film equation in plasma physics: If l(s) = (1 + s) 1 2 , we get the equation: which models the self-channeling of a high-power ultrashort laser in matter. Problem (4) has been studied extensively recently. The existence of a positive ground state solution of problem (4) has been proved by Poppenberg et al. [26] and Liu and Wang [22] by using a constrained minimization argument, which gives a solution to the equation with an unknown Lagrange multiplier λ in front of the nonlinear term. In [21], by utilizing the Nehari method, Liu et al. treated more general quasilinear problems and obtained positive and sign-changing solutions. By a change of variables in [20], the quasilinear problem was transformed to a semilinear one and an Orlitz space framework was used as the working space, and a positive solution of problem (4) was obtained by using the Mountain Pass Lemma [1]. This argument was also used later in [7], but the usual Sobolev space H 1 (R N ) framework was used as the working space. Following the idea in [7,20] for subcritical problems, the authors of [3,29] took this transformation for (4) with critical growth to obtain a positive solution. For more results about the existence of positive solutions for (4) with critical growth, we can refer to [18,23,24,25] and so on.
A natural question is wether there is a unified approach to study (3) with general functions l(s)? To answer this question, we denote g 2 (u) = 1 + (l(u 2 ) ) 2 2 , then problem (3) can be reduced to quasilinear elliptic equations (see [28]): To find solitary wave solutions of (2), it is sufficient to find positive solutions of quasilinear elliptic equations (6). By introducing a new variable replacement as follows Shen and Wang in [28] studied the existence of positive solitary wave solutions for (2) with a general functions l(s). The existence of positive solutions for problem (6) is obtained under some assumptions on g, V and h with h is subcritical growth. By using the same change of variables and variational argument, the first author, Peng and Yan studied the generalized problem (6) with critical growth, and obtained the existence of positive solutions in [9,10]. More precisely, the authors in [10] found that the critical exponents for problem (6) with general g(u) are α2 * if lim t→+∞ g(t) t α−1 = β > 0 for some α ≥ 1. Using this fact, they proposed the critical problem for given g(u) as follows: N −2 and 2α < p < α2 * . As a corollary, in [10], they established the existence of positive solutions for problem (8) if g satisfies the following assumption: (g 1 ) g ∈ C 1 (R) is an even positive function and g (t) ≥ 0 for all t ≥ 0, g(0) = 1.
Moreover, there exist some constants α ≥ 1, β > 0 and γ ∈ (−∞, α) such that It is interesting to note that α2 * = 2αN N −2 behaves like a critical exponent for equation (8). Indeed, by using a Pohozaev type variational identity they deduced that the equation has no positive solution in [10]).
We observe that formally (1) is the Euler-Lagrange equation associated with the nature energy functional It should be pointed out that we can not apply variational methods directly because the lack of an appropriate function space. We can see that the above functional is well defined in H 1 (R N ) for N ≥ 3, but it doesn't posses both smooth and compactness properties in this Sobolev space. Making use of the change of variables (7) with ,

YINBIN DENG AND WENTAO HUANG
we obtain that Since g(t) is a nondecreasing bounded function, we can deduce that 1 If u is a nontrivial solution of (1), then it should satisfy ψ, then it can be checked that (13) is equivalent to Therefore, in order to find nontrivial solutions of (1), it suffices to study the following equation We assume that the potential V ∈ C 1 (R N , R) satisfies the following conditions: Our main result can be stated as follows.
Remark 1. (i) It was shown in [11] by using a Pohozaev type identity that (5) has no positive solution in It is a interesting issue whether q * can be exactly 2 * or not.
(ii) There are indeed functions which satisfy (V 1 ) − (V 2 ). An example is given by Remark 2. Since we are going to discuss the existence of positive solutions of problem (15), we rewrite the corresponding variational functional J(v) in the following form: where v + = max{v, 0}. We claim that all nontrivial critical points of J are the positive solutions of (15). In fact, let v ∈ H 1 (R N ) be a nontrivial critical point of J, then v must be a nontrivial solution of (16) Standard regularity argument show that v ∈ C 2 (R N ). Moreover, it is easy to check that the right hand side of (16) is nonnegative. Therefore, we know from the strong maximum principle that v is positive.
In order to prove Theorem 1.1, we have to solve two difficulties. Firstly, to deal with the difficulty caused by the lack of compactness due to the nonlinearity with the critical growth, we should estimate precisely the mountain pass value. Secondly, as we have seen in [11] and [31], the effect of change of variables (7) with , which prevent us from using the standard way to prove the boundedness of (P S) sequence for p ∈ (2, 12 − 4 √ 6 ]. To overcome these difficulties, we use an abstract result developed by Jeanjean in [14], where the author studied the problem of the form when f (x, t) is asymptotically linear in t and periodic in x i , 1 ≤ i ≤ N . That is, we introduce a family of C 1 -functionals defined as where c λ is given in Lemma 3.1 below. To prove the convergence of bounded (P S) c λ sequence for J λ and obtain a nontrivial critical point v λ of J λ with J λ (v λ ) = c λ for a.e. λ ∈ [1, 2], we need to establish a version of global compactness lemma related to the functional J λ and its limiting functional we can prove that {v λm } is a bounded (P S) sequence for the original functional J = J 1 satisfying lim m→∞ J(v λm ) = c 1 and v λm → 0, which will yields Theorem 1.1.

YINBIN DENG AND WENTAO HUANG
From the assumption of (V 1 ), we can introduce an equivalent norm of H 1 (R N ) defined as The outline of this paper is as follows. In Section 2, we provide some useful lemmas and some results on least energy solutions for autonomous problems which are crucial to insure the compactness of bounded (P S) sequence. In Section 3, we employ the change of variables and an abstract result developed by Jeanjean in [14] to prove Theorem 1.1.
2. Some preliminary lemmas. In this section, we give some preliminary lemmas. First, we collect some properties of the change of variables G −1 (s).
The function G −1 (s) enjoys the following properties: By a standard argument (see for example [2]), we can obtain the following Pohozaev type identity.
Let v be a weak solution of problem (15) which equivalent to (1) and 2 < p < 2 * , then we have the following Pohozaev identity: Using the Brezis-Lieb lemma in [5], we can prove the following lemma.
On the existence of bounded (P S) sequences, we shall introduce the following abstract result developed by Jeanjean [14]. Lemma 2.4. Let X be a Banach space equipped · , and let L ⊂ R + be an interval. We consider a family (I λ ) λ∈L of C 1 -functionals on X of the form where B(u) ≥ 0, ∀u ∈ X, and such that either A(u) → +∞ or B(u) → +∞ as u → ∞. We assume that there are two points (v 1 , v 2 ) in X, such that setting Then, for almost every λ ∈ L, there is a bounded (P S) c λ sequence in X. Moreover, the map λ → c λ is continuous from the right.
In the following, we introduce some facts about the following nonlinear scalar field equation in R N with critical growth: where k : R → R is continuous function. The solution ω(x) is said to be a least energy solution (or ground state) of (19) and I is the natural energy functional corresponding to (19): where K(s) = s 0 k(t)dt. The following lemma can be regarded as a form of generalization of Berestycki and Lions [2] about the subcritical case to the critical case for N ≥ 3, we can find the details in Zhang and Zou [32].
From now on, we consider the autonomous problem related to (15) −∆v and the corresponding variational functional is Define . As a consequence of Lemma 2.5, we can obtain the following result.
The result follows after verifying that the function k(s) satisfies the assumptions (k 1 ) − (k 4 ) presented in Lemma 2.5. Indeed, the fact that (k 1 ) holds is trivial. By Finally, from the fact that 1 ≤ g(t) ≤ 3 2 and Lemma 2.1 (1), we know that Therefore, we can verify that k satisfies the assumptions (k 1 ) − (k 4 ) if we take a = V ∞ and µ = ( 2 3 ) 2 * 2 . Then the proof follows directly from Lemma 2.5.
3. Proof of Theorem 1.1. In this section, we devote to the case when the potential V (x) is not constant. Set L = [1, 2], we consider a family of C 1 -functionals on where . From Lemma 2.1 and (V 1 ) we find that Now, we can verify that the functional J λ has a Mountain Pass geometry.
On the other hand, from Lemma 2.1, it is easy to check that for any ε > 0, there exists C ε > 0 such that F (v) ≤ εv 2 + C ε |v| 2 * . Then choosing ε small enough, by Sobolev inequality, we have Then we see that J λ has a strict local minimum at 0 and hence c λ > 0.
In the following, we will give an appropriate estimate on the mountain pass value c λ .
Proof. It suffices to show that there exists v ∈ H 1 (R N )\{0} such that One of the possible candidates for v is v ε = φw ε , where φ is a smooth cut-off function such that φ(x) = 1 if |x| ≤ 1, φ(x) = 0 if |x| ≥ 2 and |∇φ| ≤ 2, and It is well-known that w ε satisfies the equation −∆u = |u| 2 * −2 u in R N , i.e., w ε is the minimizer of S = inf  Then by a direct computation we get the following estimations:

YINBIN DENG AND WENTAO HUANG
We claim that there exist positive constants t 0 , t 1 > 0 such that t 0 ≤ t ε ≤ t 1 .
First we prove that t ε is bounded from below by a positive constant. Otherwise, we could find a sequence ε n → 0 such that t εn → 0. By the above estimations, up to a subsequence, we have t εn v εn → 0 in H 1 (R N ). Therefore, 0 < sup On the other hand, we have Consider the function ξ : [0, ∞) → R given by Noting that and for ε > 0 sufficiently small, we have Consequently, From (24) and (25), we only need to prove that I < 0 for small ε. By a simple computation, we know that this is true if either p ∈ (2, 2 * ) for N ≥ 4 or p ∈ (4, 6) for N = 3.

Remark 3.
We consider the limiting functional J λ,∞ associated to the functional Then it is not difficult to check that the conclusions of Lemmas 3.1 and 3.2 hold for J λ,∞ .

YINBIN DENG AND WENTAO HUANG
Then the functional J λ and J λ,∞ can be rewritten by respectively. Moreover, by Lemma 2.1, it is not difficult to check that uniformly in x ∈ R N and λ ∈ [1,2]. Motivated by the ideas in [15] or [12,30], we can establish a version of global compactness lemma related to the functional J λ and its limiting functional J λ,∞ .
Then exist a subsequence of {v n }, still denoted by {v n }, an integer l ∈ N ∪ {0}, sequence ii) |y k n | → +∞ and |y k n − y k n | → +∞ for k = k , (iii) w k = 0 and J λ,∞ (w k ) = 0 for 1 ≤ k ≤ l, where we agree that in the case l = 0, the above holds without w k and {y k n }. Proof. Since {v n } ⊂ H 1 (R N ) is bounded, we may assume, up to a subsequence, v n v in H 1 (R N ). Then J λ (v n ) → 0 implies that for any ψ ∈ C ∞ 0 (R N ) as n → ∞. By using Lebesgue Dominated Theorem, we can obtain that J λ (v) = 0. Thus, (i) holds. By Lemma 2.2, the Pohozaev identity gives that From (V 2 ) and Hardy inequality (see [13]): Therefore, from Lemma 2.1 (1), we obtain that Step 1.
. The proof of (a.1) and (b.1) are standard, which follow from the Brezis-Lieb lemma. By (30) and Lemma 2.3, we can check that Combining (a.1), (b.1), (34) and the fact that v 1 (1), which gives the item (c.1). Finally, we prove item (d.1). By elliptic estimate, we have v ∈ L ∞ (R N ). Then from Lemma 8.9 in [30], one has ∀ψ ∈ C ∞ 0 (R N ) By the similar argument of Lemma 8.1 in [30], we also have ∀ψ ∈ C ∞ 0 (R N ) On the other hand, a direct computation shows that Combining (35)-(37) and the fact that v 1 Vanishing: If σ 1 = 0, then by Lion's compactness lemma [19], Without loss of generality, we assume that . By the Sobolev inequality, we have If d > 0, we can get that From (33), we have a contradiction. Hence, d = 0, then v n → v in H 1 (R N ) and Lemma 3.3 holds with l = 0. Non-vanishing: If σ 1 > 0, then there exists a sequence {y 1 n } ⊂ R N such that Set w 1 n = v 1 n (· + y 1 n ). Then {w 1 n } is bounded in H 1 (R N ) and we may assume that w 1 n w 1 in H 1 (R N ). Since we see that w 1 = 0. Moreover, v 1 n 0 in H 1 (R N ) implies that {y 1 n } is unbounded. Hence, we may assume that |y 1 n | → +∞. We see that it is not difficult to verify that J λ,∞ (w 1 ) = 0. Moreover, we claim that Indeed, combined with the Pohozaev identity, which similar to (18) Step 2. Set v 2 n = v n − v − w 1 (· − y 1 n ). We can similarly check that (a.2) v 2 Similar to the argument in Step 1, let If vanishing occurs, by (39) and the similar argument of vanishing case in Step 1, we know that v 2 . Moreover, by (c.2), we see that J λ (v n ) + o n (1) = J λ (v) + J λ,∞ (w 1 ) and Lemma 3.3 holds with l = 1.
If non-vanishing occurs, then there exists a sequence {y 2 n } ⊂ R N and a nontrivial w 2 ∈ H 1 (R N ) such that w 2 n = v 2 n (· + y 2 n ) w 2 in H 1 (R N ). Then by (d.2), we have that J λ,∞ (w 2 ) = 0. Furthermore, v 2 n 0 in H 1 (R N ) implies that |y 2 n | → +∞ and |y 2 n − y 1 n | → +∞. Finally, we proceed by iteration. Similar to (39), if w k is a nontrivial critical point of J λ,∞ , then J λ,∞ (w k ) > 0. So there exists some finite l ∈ N such that only the vanishing case occurs in Step l. Then the lemma is proved.
On the convergence of bounded (P S) sequence {v n } for J λ , we can establish the following lemma.
which contradicts with (40). Hence, l = 0, i.e. v n → v λ in H 1 (R N ) and then v λ is a nontrivial critical point for J λ and J λ (v λ ) = c λ . As a result, we complete the proof.
Combining Lemmas 2.4 and 3.1, we deduce that for a.e. λ ∈ [1,2], there exists . Then by Lemma 3.4, we deduce that J λ has a nontrivial critical point v λ ∈ H 1 (R N ) with J λ (v λ ) = c λ for a.e. λ ∈ [1,2]. As a special case we obtain the existence of where c λm ∈ (0, λ . In order to prove Theorem 1.1, we need to show that the critical point sequence {v λm } obtained in (42) is bounded and that is a (P S) sequence for J = J 1 satisfying lim m→∞ J(v λm ) = c 1 , where J is given by (12). Then applying Lemma 3.4 again, we obtain a nontrivial critical point of J and the proof is completed.
Proof of Theorem 1.1. First, we show that the sequence {v λm } ⊂ H 1 (R N ) obtained in (42) is bounded. Since J λm (v λm ) = c λm ≤ c 1 and from Lemma 2.2, we deduce that By the similar argument to prove (33), we can obtain R N |∇v λm | 2 dx ≤ C. Next, we only need to show the boundedness of R N v 2 λm dx. In fact, recall that J λm (v λm ) = 0, we have Therefore, by Lemma 2.1, for any ε > 0 there exists C ε > 0 such that by choosing ε > 0 small enough, we obtain R N v 2 λm dx is bounded. Therefore, {v λm } is bounded in H 1 (R N ). Then we have for any ψ ∈ C ∞ 0 (R N ) Then applying Lemma 3.4 again, we obtain a nontrivial critical point v 0 ∈ H 1 (R N ) for J and J(v 0 ) = c 1 . Finally, we end this proof by showing the existence of a ground state solution for problem (15) which equivalent to (1). Let In fact, for any v satisfying J (v) = 0, by standard argument we see v ≥ ρ for some positive constant ρ. Similar to the argument to prove (33), we infer Therefore, d ≥ 0. In the following we rule out d = 0. Suppose by contradiction that {v n } be a critical point sequence of J satisfying lim n→∞ J(v n ) = 0. From (43), we have lim n→∞ R N |∇v n | 2 dx = 0. This conclusion combined with J (v n ), v n = 0, we can verify that lim n→∞ R N v 2 n dx = 0. Therefore, we obtain lim n→∞ v n = 0, a contradiction with v n ≥ ρ > 0 for all n ∈ N.
Then let {v n } ⊂ H 1 (R N ) be a sequence of nontrivial critical point of J satisfying J(v n ) → d < 1 N ( 3 2 S) N 2 . By (V 2 ) and Hardy inequality, we can similarly deduce that {v n } is bounded in H 1 (R N ), i.e., {v n } is a bounded (P S) d sequence for J. Similar to the arguments in Lemma 3.4, there exists a nontrivial w ∈ H 1 (R N ) such that J(w) = d and J (w) = 0. Moreover, Remark 2 can show that w > 0.