Keywords:semilinear Schrödinger equation; vanishing or coercive potentials
1 Introduction and statement of results
In this paper, we consider the following semilinear elliptic equation:
with the real numbers b and s satisfying
With respect to the functions V and K, we assume that
A typical example for Eq. (1.1) with V and K satisfying (A1) and (A2) is the equation
Equation (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., [1-6] and [7-10] for reference).
Under the above assumptions, Eq. (1.1) has a natural variational structure. For an open subset Ω in , let be the collection of smooth functions with a compact support set in Ω. Let E be the completion of with respect to the inner product
From assumptions (A1) and (A2), we deduce that
are two equivalent norms in the space
Then, by the Hölder and Sobolev inequalities (see, e.g., [, Theorem 1.8]), we have, for every ,
Then the functional
In a recent paper , Alves and Souto proved that the space E can be embedded compactly into if and and Φ satisfies the Palais-Smale condition consequently. Then, by using the mountain pass theorem, they obtained a nontrivial solution for Eq. (1.1). Unfortunately, when , the embedding of E into is not compact and Φ no longer satisfies the Palais-Smale condition. Therefore, the ‘standard’ variational methods fail in this case. From this point of view, 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  and ). However, 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 that 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 stating our main result, we need to give some definitions.
Our main result reads as follows.
Theorem 1.1Under assumptions (A1) and (A2), ifb, sandpsatisfy (1.3) and (1.2) and
Notations Let X be a Banach space and . We denote the Fréchet derivative of φ at u by . The Gateaux derivative of φ is denoted by , . By → we denote the strong and by ⇀ the weak convergence. For a function u, denotes the functions . The symbol denotes the Kronecker symbol:
2 An equivalent equation for Eq. (1.1)
Lemma 2.1Under the above assumptions,
From the classical Hardy inequality (see, e.g., [, Lemma 2.1]), we deduce that for every bounded domain , there exists such that, for every ,
Proof Using the spherical coordinates
By using the divergence theorem and Lemma 2.1, we get that
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).
3 The variational functional for Eq. (2.9)
The following inequality is a variant Hardy inequality.
By using the Hölder inequality, it follows that
Then we conclude that
by (3.3) and the classical Hardy inequality (see, e.g., )
Then the desired result of this lemma follows from (3.4), (3.8) and (3.9) immediately. □
This lemma implies that
By the Sobolev inequality, we have
Recall that and if . Then, by the Hardy-Sobolev inequality (see, for example, [, Lemma 3.2]), we deduce that there exists such that (3.14) still holds. Therefore, the functional
and the critical points of J are nonnegative solutions of (2.9).
4 Some minimizing problems
we deduce that the norm defined by
Lemma 4.1The infimum
It follows that
It follows that
By Lemma 4.1, we have
It follows that
Since the functionals and are invariant by translations, the same argument as the proof of [, Theorem 1.34] yields that there exists a positive minimizer for the infimum S. Moreover, 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).
It follows that
This together with (4.14) yields the result of this lemma. □
5 The Palais-Smale condition for the functional J
Recall that J is the functional defined by (3.16). By a sequence of J, we mean a sequence such that and in as , where denotes the dual space of . J is called satisfying the condition if every sequence of J contains a convergent subsequence in .
Our main result in this section reads as follows.
Theorem 5.1Under assumptions (A1) and (A2), letbe asequence ofJ. Then replacingif necessary by a subsequence, there exist a solutionof Eq. (4.10), a finite sequence, kfunctionsandksequencessatisfying:
To prove this theorem, we need some lemmas. Our proof of this theorem is inspired by the proof of [, Theorem 8.4].
Proof If , then is bounded in . In this case, the result of this lemma is obvious. If , then as . Since , by Lemma 3.2 of , the map from is compact. Therefore, for any , there exists such that
Proof Since , by Lemma 3.2 of , the map from is compact. Therefore, for any , there exists such that
And there exists depending only on ϵ such that , . By (5.1) and the Lions lemma (see, for example, [, Lemma 1.21]), we get that
One can follow the proof of [, Lemma 8.1] step by step and use Lemma 5.2 to give the proof of this lemma.
Then j is a convex function. From [, Lemma 3], we have that for any , there exists such that for all ,
By Lemma 3.2 of , the map from is compact. We get that there exists such that for any n,
The left proof is the same as the proof of [, Lemma 1.32]. □
(2) Lemma 5.5 implies
By Lemma 5.4, we have
Proof We divide the proof into several steps.
Combining the above two limits leads to
By (5.11) and the Hölder inequality, we have
It follows that
This together with (5.14), (5.15) and
Combining (5.10), (5.12) and (5.16) leads to
By (5.11), (5.20) and (5.21), we get that
Combining (5.19), (5.22) and (5.17) leads to
We obtain from Lemma 5.5 that
By Lemma 5.2,
Combining (5.24)-(5.26) yields
First, as (5.9), we have
Combining (5.31) and (5.32) yields that
Finally, combining (5.28), (5.29) and (5.34) leads to
Proof of Theorem 5.1 We divide the proof into two steps.
(1) For n big enough, we have
Let us define
it follows from the Rellich theorem that
Moreover, Lemma 4.3 implies that
contains a convergent subsequence.
6 Proof of Theorem 1.1
Recall that the critical points of J are nonnegative solutions of (2.9). By Corollary 2.2, to prove that Eq. (1.1) has a positive solution, it suffices to prove that J has a nontrivial critical point. Moreover, by Corollary 5.8, it suffices to apply the classical mountain pass theorem (see, e.g., [, Theorem 1.15]) to J with the mountain pass value .
By (3.14), we get
By Corollary 5.8 and the mountain pass theorem (see [, Theorem 1.15]), J has a critical value c such that and Eq. (2.9) has a positive solution . 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 . Using the divergence theorem, Lemma 2.1 and (2.12), we get that
Moreover, by Lemma 2.1 and (2.12), we get that
The author declares that they have no competing interests.
Shaowei Chen was supported by Science Foundation of Huaqiao University.
Pankov, AA, Pflüer, K: On semilinear Schrödinger equation with periodic potential. Nonlinear Anal.. 33, 593–609 (1998). Publisher Full Text
Su, J, Wang, Z-Q, Willem, M: Weighted Sobolev embedding with unbounded and decaying radial potentials. J. Differ. Equ.. 238, 201–219 (2007). Publisher Full Text
Sirakov, B: Existence and multiplicity of solutions of semi-linear elliptic equations in . Calc. Var. Partial Differ. Equ.. 11, 119–142 (2000). Publisher Full Text
Alves, CA, Souto, MS: Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity. J. Differ. Equ.. 254, 1977–1991 (2013). Publisher Full Text
Garcia Azorero, JP, Peral Alonso, I: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ.. 144, 441–476 (1998). Publisher Full Text
Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDES involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc.. 352, 5703–5743 (2000). Publisher Full Text