Existence of positive solutions for a critical nonlinear Schrodinger equation with vanishing or coercive potentials

In this paper we investigate the existence of the positive solutions for the following nonlinear Schr\"odinger equation $$ -\triangle u+V(x)u=K(x)|u|^{p-2}u\ {in}\ \mathbb{R}^N $$ where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as $|x|\rightarrow\infty$ with $0<a,\mu<+\infty$, $b<2,$ $ b\neq0$, $ 0<\frac{s}{b}<1$ and $p=2(N-2s/b)/(N-2).$


Introduction and statement of results
In this paper, we consider the following semilinear elliptic equation When 0 < b < 2, the potentials are vanishing at infinity and when b < 0, the potentials are coercive. Eq.(1.1) arises in various applications, such as chemotaxis, population genetics, chemical reactor theory, and the study of standing wave solutions of certain nonlinear Schrödinger equations. Therefore, they have received growing attention in recent years (one can see, e.g., [2], [3], [5], [10], [11] and [13] for reference).
Under the above assumptions, Eq.(1.1) has a natural variational structure. For an open subset Ω in R N , let C ∞ 0 (Ω) be the collection of smooth functions with compact support set in Ω. Let E be the completion of C ∞ 0 (R N ) with respect to the inner product From the assumptions (A 1 ) and (A 2 ), we deduce that are two equivalent norms in the space Therefore, there exists B 1 > 0 such that Moreover, the assumptions (A 1 ) and (A 2 ) imply that there exists B 2 > 0 such that Then by the Hölder and the Sobolev inequalities (see, e.g., [14,Theorem 1.8]), we have, for every u ∈ C ∞ 0 (R N ), where C > 0 is a constant independent of u. It follows that there exists a constant C ′ > 0 such that This implies that E can be embedded continuously into the weighted L p −space Then the functional is well defined in E. And it is easy to check that Φ is a C 2 functional and the critical points of Φ are solutions of (1.1) in E.
In a recent paper [1], Alves and Souto proved that the space E can be embedded compactly into L p K (R N ) if 0 < b < 2 and 2(N − 2s/b)/(N − 2) < p < 2 * and Φ satisfies Palais-Smale condition consequently. Then by using the mountain pass theorem, they obtained a nontrivial solution for Eq.(1.1). Unfortunately, when p = 2(N − 2s/b)/(N − 2), the embedding of E into L p K (R N ) is not compact and Φ satisfies no longer Palais-Smale condition. Therefore, the "standard" variational methods fail in this case. From this point of view, p = 2(N − 2s/b)/(N − 2) should be seen as a kind of critical exponent for Eq.(1.1). If the potentials V and K are restricted to the class of radially symmetric functions, "compactness" of such a kind is regained and "standard" variational approaches work (see [11] and [13]). But this method does not seem to apply to the more general equation (1.1) where K and V are non-radially symmetric functions.
It is not easy to deal with Eq. (1.1) directly because there are no known approaches can be used directly to overcome the difficulty brought by the loss of compactness. However, in this paper, through an interesting transformation, we find an equivalent equation for Eq. (1.1) (see Eq. (2.9) in Section 2). This equation has the advantages that its Palais-Smale sequence can be characterized precisely through the concentration-compactness principle (see Theorem 5.1) and it possesses partial compactness (see Corollary 5.8). By means of these advantages, a positive solution for this equivalent equation and then a corresponding positive solution for Eq. (1.1) are obtained.
Before to state our main result, we need to give some definitions. Let and Let H 1 (R N ) be the the Sobolev space endowed with the norm and the inner product respectively and L p (R N ) be the function space consisting of the functions on R N that are p−integrable.
We denote this infimum by S p .
Our main result reads as follows: Then for any ǫ > 0, there exist r ǫ > 0 and u ǫ ∈ H 1 0 (R r ) \ {0} such that

It follows from this inequality and Rr
Notations: Let X be a Banach Space and ϕ ∈ C 1 (X, R). We denote the Fréchet derivative of ϕ at u by ϕ ′ (u). The Gateaux derivative of ϕ is denoted by ϕ ′ (u), v , ∀u, v ∈ X. By → we denote the strong and by ⇀ the weak convergence. For a function u, u + denotes the functions max{u(x), 0}. The symbol δ ij denotes the Kronecker symbol: We use o(h) to mean o(h)/|h| → 0 as |h| → 0.

Moreover
, By using the divergence theorem and Lemma 2.1, we get that Moreover, Therefore, This completes the proof. ✷ This theorem implies that the problem of looking for solutions of (1.1) can be reduced to a problem of looking for solutions of (2.9).

The variational functional for Eq. (2.9).
The following inequality is a variant Hardy inequality.
Proof. We only give the proof of (3 By using the Hölder inequality, it follows that And then we conclude that ✷ From the definition of A ij (x) (see (2.3)), it is easy to verify that, for u ∈ H 1 (R N ), There exist constants C 1 > 0 and C 2 > 0 such that, for every u ∈ H 1 (R N ), Proof. From the conditions (A 1 ) and (A 2 ), we deduce that there exists a constant C > 0 such that 3) and the classical Hardy inequality (see, e.g., [7]) we deduce that there exists a constant C > 0 such that This together with the fact that R N |x·∇u| 2 In this case, The conditions (A 1 ) and (A 2 ) imply that there exists a constant C > 0 such that Combining (3.5) − (3.7) yields that there exists a constant C 1 > 0 such that If b < 0, (3.7) still holds. From Lemma 3.1 and (3.7), we deduce that there exists a constant C 1 > 0 such that, for every u ∈ H 1 (R N ), Then the desired result of this lemma follows from (3.4), (3.8) and (3.9) immediately. ✷ This lemma implies that is equivalent to the standard norm || · || in H 1 (R N ). We denote the inner product associated with || · || A by (·, ·) A , i.e., By the Sobolev inequality, we have and By the condition (A 1 ) and (A 2 ), if 0 < b < 2, then K * is bounded in R N . Therefore, by (3.13), there exists C > 0 such that However, if b < 0, K * has a singularity at x = 0, i.e., Then by the Hardy-Sobolev inequality (see, for example, [8, Lemma 3.2]), we deduce that there exists C > 0 such that (3.14) still holds. Therefore, the functional is a C 2 functional defined in H 1 (R N ). Moreover, it easy to check that the Gateaux derivative of J is and the critical points of J are nonnegative solutions of (2.9).
Proof. In this proof, we always view a vector in R N as a 1 × N matrix. And we use A T to denote the conjugate matrix of a matrix A.

It follows that
By (4.6) and θ ′ · G T = θ, we get that

It follows that
✷ Since the functionals ||u|| 2 θ and R N |u| p dx are invariant by translations, the same argument as the proof of [14, Theorem 1.34] yields that there exists a positive minimizer U θ for the infimum S. And from the Lagrange multiplier rule, it is a solution of In the next section, we shall show that Eq.(4.9) is the "limit" equation of It is easy to verify that and the critical points of this functional are solutions of (4.9).
Proof. Since u is a critical point of J θ , we have It follows that (4.14) Since u = 0, by ||u|| 2 θ = µ R N (u + ) p dx and ||u|| 2 θ ≥ S( R N (u + ) p dx) 2/p , we get that This together with (4.14) yields the result of this lemma. ✷

Palais-Smale conditions for the functional J.
Recall that J is the functional defined by (3.16). By a (P S) c sequence of J, we mean a sequence {u n } ⊂ . |y l n | → ∞, |y l n − y l ′ n | → ∞, l = l ′ , n → ∞, This theorem gives a precise representation of (P S) c sequence for the functional J. Through it, partial compactness for J can be regained (see Corollary 5.8).
To prove this theorem, we need some lemmas. Our proof of this theorem is inspired by the proof of [14,Theorem 8.4]. If |y n | → ∞, n → ∞, then In this case, the result of this lemma is obvious.
Since y n → ∞, we have lim K * (x + y n ) = µ. Then using the Lebesgue theorem and the above two inequalities, we get that Let ǫ → 0. Then we get the desired result of this lemma. ✷ And there exists D(ǫ) > 0 depending only on ǫ such that K * (x) ≤ D(ǫ), |x| ≥ δ ǫ . By (5.1) and the Lions Lemma (see, for example, [14, Lemma 1.21]), we get that One can follow the proof of [14, Lemma 8.1] step by step and use Lemma 5.2 to give the proof of this lemma.
The following Lemma is a variant Brézis-Lieb Lemma (see [4]) and its proof is similar to that of [14,Lemma 1.32 ].
. Then j is a convex function. From [4, Lemma 3 ], we have for any By Lemma 3.2 of [8], the map v → K 1/p * v from H 1 (R N ) → L p loc (R N ) is compact. We get that there exists δ ǫ > 0 such that for any n, |x+yn|<δǫ f ǫ n dx < ǫ.

2). For any
By the definition of the inner product (·, ·) A (see (3.11)), we have Since u n (· + y n ) ⇀ u in H 1 (R N ), we have Moreover, together with (2.8) and the fact that |y n | → ∞ yields that for any fixed R > 0 Combining the above two limits leads to By (5.11) and the Hölder inequality, we have Since ∇h ∈ L 2 (R N ), for any ǫ > 0, there exists R ǫ > 0 such that
Moreover, there exists t 0 > 0 such that ||t 0 u 0 || A > r and J(t 0 u 0 ) < 0. It follows from (6.1) that By Corollary 5.8 and the mountain pass theorem (see [14,Theorem 1.15]), J has a critical value c such that b ≤ c < ( 1 2 − 1 p )µ − 2 p−2 S p p−2 and Eq.(2.9) has a positive solution v ∈ H 1 (R N ). Then by Theorem 2.2, the function u defined by (2.1) is a positive solution of (1.1). To complete the proof, it suffices to prove that u ∈ E. Using the divergence theorem, Lemma 2.1 and (2.12), we get that Moreover, by Lemma 2.1 and (2.12), we get that