- Open Access
Solutions of semiclassical states for perturbed p-Laplacian equation with critical exponent
© Wang et al.; licensee Springer. 2014
- Received: 25 June 2014
- Accepted: 4 November 2014
- Published: 19 November 2014
In this paper, we study semiclassical states for perturbed p-Laplacian equations. Under some given conditions and minimax methods, we show that this problem has at least one positive solution provided that ; for any , it has m pairs of solutions if , where ℰ, are sufficiently small positive numbers. Moreover, these solutions in as .
- semiclassical states
- positive solutions
- critical exponent
where , is the p-Laplacian operator with , , is the Sobolev critical exponent, is a nonnegative potential, is bounded positive coefficient, and is a p-superlinear but subcritical function.
where , is a given potential, κ, ħ are real constants and ρ, are real functions. The quasilinear equation (1.2) appears more naturally in mathematical physics and has been derived as models of several physical phenomena corresponding to various types of . In the case , (1.2) models the superfluid film equation in fluid mechanics by Kurihara . In the case , (1.2) models the self-channeling of a high-power ultra short laser in matter (see –). For more physical motivations and more references dealing with applications, we can refer to – and references therein.
with , and .
When , , , , we can refer to , –, and so on. Here positive or sign-changing solutions were obtained by using a constrained minimization argument, or a Nehari method, or a technique of changing variables. We remark that among the above three methods, the last one, which was first proposed in , is most effective for the power nonlinearity case since this argument can transform the quasilinear problem to a semilinear one and an Orlicz space framework was used as the working space.
It is worth pointing out that the critical exponent case was mentioned as an open problem in , where the authors observed that the number 22∗ behaves like a critical exponent for (1.3). In , for , the authors treated the case where the nonlinearity has critical exponential growth, that is, h behaves like as . For , when satisfies radially symmetrical, periodic, and some geometric conditions, Moameni  obtained the existence of nonnegative solutions for (1.3) with the critical growth case; when satisfied asymptotic and periodic condition. In , , the authors prove the existence of ground state solutions for (1.3) with or . In the present paper, we will consider a class of quasilinear Schrödinger equations with a nonperiodic potential function in , . In fact, we will investigate the existence of solutions for the critical growth case when the parameter ε goes to zero, i.e., the semiclassical problems for the critical quasilinear Schrödinger equation (1.1). It is well known that in this case the laws of quantum mechanics must reduce to those of classical mechanics, and it describes the transition between quantum mechanics and classical mechanics. As far as we know, there are few papers considering the existence and concentration of semiclassical states for quasilinear Schrödinger equations. For instance, in , , using a suitable Trudinger-Moser inequality in and a penalization technique, the authors established the existence of semiclassical solutions for the critical exponent case via the mountain pass lemma.
Inspired by , we will extend the existence and multiplicity of solutions for (1.5) to the general case for (1.5) with , . Moreover, the corresponding problem becomes more complicated: first, is not a Hilbert space when ; secondly, the weak continuity of operator in is difficulty to establish.
In this paper, we make the following assumptions:
(V1): and there is such that the set has finite Lebesgue measure.
(h1):, , uniformly in x as .
(h3): There are , , such that and .
A typical example satisfying (h1)-(h3) is the function with and being positive and bounded.
Our main results of this paper are as follows.
Assume that (V1)-(V2), (K), and (h1)-(h3) hold, and. Then for anyandthere issuch that if, problem (1.1) has at least m pairs of solutions, , , which satisfy the estimates (i) and (ii) in Theorem 1.1. Moreover, inas.
These results are new for the p-Laplacian equation and are a generalization of the results in .
Because the nonhomogeneous term prevents us from working directly with the functional , which is not well defined in since, for , may hold. The other difficulty is the lack of compactness due to the unboundedness of the domain and the appearance of the Sobolev critical exponent . To overcome these difficulties we generalize an argument developed by Liu et al. in  for , (see also ). We make the change of variables , and reformulate the problem into a new one which has an associated functional that is well defined and is of class on .
Before we end this section, some notations are in order. We use to denote the integral , denotes the usual norm . In the whole paper, C denotes a generic constant, which may vary from line to line.
The rest of this paper is organized as follows: in Section 2, we describe the analytic setting where we restate the problems in equivalent form by replacing with other than the usual scaling (see ), due to the non-autonomy of nonlinearities. In Section 3, we show that the corresponding energy functional satisfies the (PS) condition at the levels less than with some independent of λ. Thus in Section 4 we construct minimax levels less than for all λ large enough. We prove our main results in Section 5.
Thus we collect some properties of f.
f is uniquely defined function and invertible.
for all .
for all .
for all .
for all .
- (8)There exists a positive constant C such that
and the first inequality in (6) is proved. The second inequality in (6) is obtained in a similar way.
Thus is a nondecreasing function for and this together with estimate (5) shows item (7). Point (8) is an immediate consequence of (4) and (7). Point (9) is obtained from the definition of f. □
Now we can restate Theorem 1.1 and Theorem 1.2 as follows.
Let (V1)-(V2), (K), and (h1)-(h3) hold, and. Then for anyandthere issuch that if, problem (2.3) has at least m pairs of solutions, , , which satisfy the estimates (i) and (ii) in Theorem 2.2. Moreover, inas.
where , then and critical points of are positive solutions for (2.3).
Let E be a real Banach space and be a function of class . We say that is a (PS) c sequence if and . is said to satisfy the (PS) c condition if any (PS) c sequence contains a convergent subsequence.
The main result of the section is the following compactness result.
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Letbe a (PS) c sequence for. Thenandis bounded in E.
where as .
Taking the limit in (3.2), we can obtain .
In the following, we need to show is bounded in E. From (3.3), we need to prove that is bounded.
These estimates imply that is bounded in E. □
From Lemma 3.1, we know that every (PS) c sequence is bounded, hence, without loss of generality, we may assume in E and , in for , and a.e. for . Obviously, v is a critical point of .
for all, where.
for all .
For , we only need Lemma 2.1. □
From the proof of Lemma 3.2, we can find the same subsequence such that the result of Lemma 3.2 holds for both and .
uniformly in .
which implies the conclusion as required. □
These, together with the facts and as , give conclusion (i).
uniformly in , proving (ii). □
then , by (3.5), if and only if . Assume that has no convergent subsequence. Then . By Lemma 3.5, one also has a subsequence that and .
The proof is complete. □
From Lemma 3.6, we have the following conclusions.
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Thensatisfies the (PS) c condition for all.
Assume that (V1)-(V2), (K), and (h1)-(h3) are satisfied. Thensatisfies the (PS) c condition for all.
and there exists such that ;
there exists such that .
From now on, we consider , and the following lemma implies that possesses the mountain pass geometry.
and there exists such that .
- (ii)For any finite-dimensional subspace ,
- (iii)For any there exists such that, for each , there is such that and
- (i)First note that, for each λ, . Now, for every , define
- (ii)Observe that, by (h3), . Define the functional by
From Lemma 4.1 and Lemma 4.2(i)-(ii), if satisfies the (PS) c condition for all , then Theorem 2.2 follows from a variant mountain pass theorem. However, in general we do not know if satisfies the (PS) c condition. By Lemma 3.7 for λ large and small enough, satisfies the (PS) condition. Thus we will find a special finite-dimensional subspace by which we construct sufficiently small minimax levels for when λ is large enough.
For any , one can choose nonnegative such that the function defined by (4.5) is nonnegative. In fact, if is a sequence in with and , then by Kato’s inequality, the absolute value sequence with and , where denotes the set of all continuous functions in with compact supports. Therefore, Lemma 4.2 is still true with the function .
As a consequence of Lemma 4.2 and Remark 4.3, we have the following conclusions.
In section, we prove the existence and multiplicity results.
Proof of Theorem 2.2
where . Lemma 3.7 implies that satisfies the (PS) condition, thus there is such that and , then is a positive solution of (2.1). Moreover, it is well known that a mountain pass solution is a state solution of (2.1).
which means in as . The proof is completed. □
By the same arguments as applied to , we can obtain the existence of positive solutions for (2.3).
If is finite and satisfies the (PS) condition, then we know all are critical values for .
Proof of Theorem 2.2
Consider the functional , from (h1)-(h3), we know, for each λ, there is a closed subset of E and such that .
Thus, for any and , there exists such that , we can choose an m-dimensional subspace with .
where defined by (5.1).
The authors would like to thank the referees for valuable comments and suggestions on improving this paper. This research is supported by NSFC: Tianyuan Foundation (11326145, 11326139), and also supported by Hubei Provincial Department of Education (Q20122504) and Youth Science Foundation program of Jiangxi Provincial (20142BAB211010).
- Kurihara S: Large-amplitude quasi-solitons in superfluid films. J. Phys. Soc. Jpn. 1981, 50: 3262-3267. 10.1143/JPSJ.50.3262MathSciNetView ArticleGoogle Scholar
- Brandi HS, Manus C, Mainfray G, Lehner T, Bonnaud G: Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. I. Paraxial approximation. Phys. Fluids B 1993, 5: 3539-3550. 10.1063/1.860828View ArticleGoogle Scholar
- Chen XL, Sudan RN: Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma. Phys. Rev. Lett. 1993, 70: 2082-2085. 10.1103/PhysRevLett.70.2082View ArticleGoogle Scholar
- De Bouard A, Hayashi N, Saut J: Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun. Math. Phys. 1997, 189: 73-105. 10.1007/s002200050191MathSciNetView ArticleGoogle Scholar
- Ritchie B: Relativistic self-focusing and channel formation in laser-plasma interactions. Phys. Rev. E 1994, 50: 687-689. 10.1103/PhysRevE.50.R687View ArticleGoogle Scholar
- Bass FG, Nasanov NN: Nonlinear electromagnetic-spin waves. Phys. Rep. 1990, 189: 165-223. 10.1016/0370-1573(90)90093-HView ArticleGoogle Scholar
- Brüll L, Lange H: Solitary waves for quasilinear Schrödinger equations. Expo. Math. 1986, 4: 279-288.Google Scholar
- Kosevich AM, Ivanov BA, Kovalev AS: Magnetic solitons. Phys. Rep. 1990, 194: 117-238. 10.1016/0370-1573(90)90130-TView ArticleGoogle Scholar
- Poppenberg M, Schmitt K, Wang Z: On the existence of soliton solutions to quasilinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 2002, 14: 329-344. 10.1007/s005260100105MathSciNetView ArticleGoogle Scholar
- Quispel GRW, Capel HW: Equation of motion for the Heisenberg spin chain. Physica A 1982, 110: 41-80. 10.1016/0378-4371(82)90104-2MathSciNetView ArticleGoogle Scholar
- Ambrosetti A, Badiale M, Cingolani S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 1997, 140: 285-300. 10.1007/s002050050067MathSciNetView ArticleGoogle Scholar
- del Pino M, Felmer P: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1998, 15: 127-149. 10.1016/S0294-1449(97)89296-7MathSciNetView ArticleGoogle Scholar
- del Pino M, Felmer P: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 2002, 324: 1-32. 10.1007/s002080200327MathSciNetView ArticleGoogle Scholar
- Floer A, Weinstein A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0MathSciNetView ArticleGoogle Scholar
- Grossi M: Some results on a class of nonlinear Schrödinger equations. Math. Z. 2000, 235: 687-705. 10.1007/s002090000158MathSciNetView ArticleGoogle Scholar
- Gui C: Existence of multi-bumb solutions for nonlinear Schrödinger equations via variational method. Commun. Partial Differ. Equ. 1996, 21: 787-820. 10.1080/03605309608821208MathSciNetView ArticleGoogle Scholar
- Jeanjean L, Tanaka K: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 2004, 21: 287-318. 10.1007/s00526-003-0261-6MathSciNetView ArticleGoogle Scholar
- Kang X, Wei J: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Equ. 2000, 5: 899-928.MathSciNetGoogle Scholar
- Li Y: On a singularly perturbed elliptic equation. Adv. Differ. Equ. 1997, 2: 955-980.Google Scholar
- Oh Y: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V) α . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleGoogle Scholar
- Oh Y: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 1990, 131: 223-253. 10.1007/BF02161413View ArticleGoogle Scholar
- Pistoia A: Multi-peak solutions for a class of nonlinear Schrödinger equations. NoDEA Nonlinear Differ. Equ. Appl. 2002, 9: 69-91. 10.1007/s00030-002-8119-8MathSciNetView ArticleGoogle Scholar
- Rabinowitz PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992, 43: 270-291. 10.1007/BF00946631MathSciNetView ArticleGoogle Scholar
- Wang X: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys. 1993, 153: 229-244. 10.1007/BF02096642View ArticleGoogle Scholar
- Colin M, Jeanjean L: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal. TMA 2004, 56: 213-226. 10.1016/j.na.2003.09.008MathSciNetView ArticleGoogle Scholar
- Liu J, Wang Z: Soliton solutions for quasilinear Schrödinger equations. I. Proc. Am. Math. Soc. 2003, 131: 441-448. 10.1090/S0002-9939-02-06783-7View ArticleGoogle Scholar
- Liu J, Wang Z: Symmetric solutions to a modified nonlinear Schrödinger equation. Nonlinearity 2008, 21: 121-133. 10.1088/0951-7715/21/1/007MathSciNetView ArticleGoogle Scholar
- Liu J, Wang Y, Wang Z: Soliton solutions for quasilinear Schrödinger equations. II. J. Differ. Equ. 2003, 187: 473-493. 10.1016/S0022-0396(02)00064-5View ArticleGoogle Scholar
- Liu J, Wang Y, Wang Z: Solutions for quasilinear Schrödinger equations via the Nehari method. Commun. Partial Differ. Equ. 2004, 29: 879-901. 10.1081/PDE-120037335View ArticleGoogle Scholar
- do Ó JM, Miyagaki O, Soares S: Soliton solutions for quasilinear Schrödinger equations: the critical exponential case. Nonlinear Anal. 2007, 67: 3357-3372. 10.1016/j.na.2006.10.018MathSciNetView ArticleGoogle Scholar
- Moameni A:Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in . J. Differ. Equ. 2006, 229: 570-587. 10.1016/j.jde.2006.07.001MathSciNetView ArticleGoogle Scholar
- do Ó JM, Miyagaki O, Soares S: Soliton solutions for quasilinear Schrödinger equations with critical growth. J. Differ. Equ. 2010, 248: 722-744. 10.1016/j.jde.2009.11.030View ArticleGoogle Scholar
- do Ó JM, Severo U: Solitary waves for a class of quasilinear Schrödinger equations in dimension two. Calc. Var. Partial Differ. Equ. 2010, 38: 275-315. 10.1007/s00526-009-0286-6View ArticleGoogle Scholar
- do Ó JM, Moameni A, Severo U: Semi-classical states for quasilinear Schrödinger equations arising in plasma physics. Commun. Contemp. Math. 2009, 11: 547-583. 10.1142/S021919970900348X