- Open Access
Existence of nontrivial solutions for perturbed p-Laplacian system involving critical nonlinearity and magnetic fields
© Zhang et al.; licensee Springer. 2013
- Received: 5 October 2012
- Accepted: 9 January 2013
- Published: 24 January 2013
Under the suitable assumptions, we establish the existence of nontrivial solutions for a perturbed p-Laplacian system in with critical nonlinearity and magnetic fields by using the variational method.
MSC:35B33, 35J60, 35J65.
- p-Laplacian system
- critical nonlinearity
- magnetic fields
- variational method
where , i is the imaginary unit, is real vector potential, , is a non-negative potential, denotes the Sobolev critical exponent for and is a bounded positive coefficient.
Then solves (1.3) if and only if solves (1.2) with and .
There are also many works dealing with the magnetic fields and for the scalar case corresponding to (1.1). In , the authors firstly obtained the existence of standing waves for special classes of magnetic fields. For many results, we refer the reader to [17–22].
where f is a subcritical nonlinearity without some growth conditions such as the Ambrosetti-Rabinowitz condition. The problem (1.4) has also been studied in [28–32]. The main difficulty in treating this class of equation (1.4) is a possible lack of compactness due to the unboundedness of the domain.
However, to our best knowledge, it seems there is almost no work on the existence of non-trivial solutions to the problem (1.1) involving critical nonlinearity and magnetic fields. We mainly follow the idea of . Observe that though the idea was used in other problems, the adaption of the procedure to the problem is not trivial at all. Because of the appearance of magnetic fields , we must deal with the problem for complex-valued functions and therefore we need more delicate estimates.
The outline of the paper is as follows. The forthcoming section is the main result and preliminary results including the appropriate space setting to work with. In Section 3, we study the behavior of sequence. Section 4 gets that the functional associated to the problem possesses the mountain geometry structure, and the last section concludes the proof of the main result.
Firstly, we make the following assumptions on , , and throughout the paper:
() , and there exists such that the set has finite Lebesgue measure;
() and ;
() , ;
() and as ;
() there are , and such that and .
Under the above mentioned conditions, we get the following result.
We are going to prove the following result.
Set and for any .
(the bar denotes complex conjugation). This inequality shows that if , then and therefore for any . That is to say, if in , then in for any and a.e. in .
We call a sequence a sequence if and strongly in ( is the dual space of E). is said to satisfy the condition if any sequence contains a convergent subsequence.
The main result of Section 3 is the following compactness result.
Proposition 3.1 Let the assumptions of Theorem 2 be satisfied. There exists a constant independent of λ such that, for any sequence for with , either or .
As a consequence, we obtain the following result.
Proposition 3.2 Assume that the assumptions of Proposition 3.1 hold, satisfies the condition for all .
In order to prove Proposition 3.1, we need the following lemmas.
Lemma 3.1 Let the assumptions of Theorem 2 be satisfied. is a sequence of . Then and is bounded in the space E.
Then is bounded and . □
From Lemma 3.1, we may assume in E and in for any and , a.e. in .
Proof The proof of Lemma 3.2 is similar to that of Lemma 3.2 of , so we omit it. □
uniformly in with .
Let , , then , . From (3.1), we get in E if and only if in E.
Now, we consider the energy level of the functional below which the condition holds.
This completes the proof of Proposition 3.1. □
In connection with and Proposition 3.1, we complete this proof. □
The fact implies the desired conclusion. □
Since all norms in a finite-dimensional space are equivalent and , we complete the proof. □
In the following, we will find special finite-dimensional subspaces by which we establish sufficiently small mini-max levels.
Obviously, it follows that and for all .
Then, for any , there are with and such that .
where is defined in Lemma 4.1.
Proof This proof is similar to that of Lemma 4.3 in , so we omit the details. □
By Proposition 3.1, we know that satisfies the condition. Hence, by the mountain-pass theorem, there is such that and . This shows is a weak solution of (2.1).
Furthermore, together with the diamagnetic inequality, we prove that satisfies the estimate (2.2). The proof is complete. □
The authors would like to appreciate the referees for their precious comments and suggestions about the original manuscript. This research is supported by the National Natural Science Foundation of China (11271364) and the Fundamental Research Funds for the Central Universities (2012QNA46).
- Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar
- Benci V: On critical point theory of indefinite functions in the presence of symmetries. Trans. Am. Math. Soc. 1982, 274: 533-572. 10.1090/S0002-9947-1982-0675067-XMathSciNetView ArticleGoogle Scholar
- Brézis H, Nirenberg L: Positive solutions of nonlinear elliptic equation involving critical Sobolev exponents. Commun. Pure Appl. Math. 1983, 16: 437-477.View ArticleGoogle Scholar
- Cingolani S, Nolasco M: Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equation. Proc. R. Soc. Edinb. 1998, 128: 1249-1260. 10.1017/S030821050002730XMathSciNetView ArticleGoogle Scholar
- Del Pino M, Felmer PL: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 1997, 149: 245-265. 10.1006/jfan.1996.3085MathSciNetView ArticleGoogle Scholar
- Del Pino M, Felmer PL: 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
- Ding YH, Lin FH: Solutions of perturbed Schrödinger equations with critical nonlinearity. Calc. Var. 2007, 30: 231-249. 10.1007/s00526-007-0091-zMathSciNetView ArticleGoogle Scholar
- Guedda M, Veron L: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 1989, 12: 879-902.MathSciNetView 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.MathSciNetView ArticleGoogle Scholar
- Kang X, Wei JC: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Equ. 2000, 5: 899-928.MathSciNetGoogle Scholar
- Li YY: On a singularly perturbed elliptic equation. Adv. Differ. Equ. 1997, 2: 955-980.Google Scholar
- Oh YG: 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
- Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleGoogle Scholar
- Pistoia A: Multi-peak solutions for a class of some results on a class of nonlinear Schrödinger equations. Nonlinear Differ. Equ. Appl. 2002, 9: 69-91. 10.1007/s00030-002-8119-8MathSciNetView 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
- Esteban M, Lions PL: Stationary solutions of nonlinear Schrödinger equation with an external magnetic field. In Partial Differential Equations and the Calculus of Variations, Essays in Honor of E. De Giorgi. Brikhäuser, Basel; 1990:369-408.Google Scholar
- Arioli G, Szulkin A: A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Ration. Mech. Anal. 2003, 170: 277-295. 10.1007/s00205-003-0274-5MathSciNetView ArticleGoogle Scholar
- Bartsch T, Dancer EN, Peng S: On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Differ. Equ. 2006, 11: 781-812.MathSciNetGoogle Scholar
- Cao D, Tang Z: Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields. J. Differ. Equ. 2006, 222: 381-424. 10.1016/j.jde.2005.06.027MathSciNetView ArticleGoogle Scholar
- Cingolani S: Semiclassical stationary states of nonlinear Schrödinger equation with an external magnetic field. J. Differ. Equ. 2003, 188: 52-79. 10.1016/S0022-0396(02)00058-XMathSciNetView ArticleGoogle Scholar
- Han P: Solutions for singular critical growth Schrödinger equation with magnetic field. Port. Math. 2006, 63: 37-45.MathSciNetGoogle Scholar
- Kurata K: Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal. 2000, 41: 763-778. 10.1016/S0362-546X(98)00308-3MathSciNetView ArticleGoogle Scholar
- Alves CO, Ding YH: Multiplicity of positive solutions to a p -Laplacian equation involving critical nonlinearity. J. Math. Anal. Appl. 2003, 279: 508-521. 10.1016/S0022-247X(03)00026-XMathSciNetView ArticleGoogle Scholar
- Gloss E: Existence and concentration of bound states for a p -Laplacian equation in . Adv. Nonlinear Stud. 2010, 10: 273-296.MathSciNetGoogle Scholar
- Liu CG, Zheng YQ: Existence of nontrivial solutions for p -Laplacian equations in . J. Math. Anal. Appl. 2011, 380: 669-679. 10.1016/j.jmaa.2011.02.064MathSciNetView ArticleGoogle Scholar
- Manásevich R, Mawhin J: Boundary value problems for nonlinear perturbations of vector p -Laplacian-like operators. J. Korean Math. Soc. 2000, 5: 665-685.Google Scholar
- Zhang HX, Liu WB: Existence of nontrivial solutions to perturbed p -Laplacian system in involving critical nonlinearity. Bound. Value Probl. 2012., 2012: Article ID 53Google Scholar
- Dinu TL: Entire solutions of Schrödinger systems with discontinuous nonlinearity and sign-changing potential. Math. Model. Anal. 2006, 13(3):229-242.MathSciNetGoogle Scholar
- Dinu TL: Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev space with variable exponent. Nonlinear Anal. 2006, 65(7):1414-1424. 10.1016/j.na.2005.10.022MathSciNetView ArticleGoogle Scholar
- Gazzola F, Radulescu V: A nonsmooth critical point theory approach to some nonlinear elliptic equations in unbounded domains. Differ. Integral Equ. 2000, 13: 47-60.MathSciNetGoogle Scholar
- Ghanmi A, Maagli H, Radulescu V, Zeddini N: Large and bounded solutions for a class of nonlinear Schrödinger stationary systems. Anal. Appl. 2009, 7: 391-404. 10.1142/S0219530509001463MathSciNetView ArticleGoogle Scholar
- Rabinowitz PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992, 43: 270-291. 10.1007/BF00946631MathSciNetView ArticleGoogle Scholar
- Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.View ArticleGoogle Scholar
- Ghoussoub N, Yuan C: Multiple solutions for quasi-linear PDES involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 2000, 352: 5703-5743. 10.1090/S0002-9947-00-02560-5MathSciNetView ArticleGoogle Scholar
- Li YY, Guo QQ, Niu PC: Global compactness results for quasilinear elliptic problems with combined critical Sobolev-Hardy terms. Nonlinear Anal. 2011, 74: 1445-1464. 10.1016/j.na.2010.10.018MathSciNetView ArticleGoogle Scholar
- Brézis H, Lieb E: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc. 1983, 88: 486-490.View ArticleGoogle Scholar
- Pucci P, Radulescu V: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital., Ser. IX 2010, 3: 543-584.MathSciNetGoogle Scholar
- Radulescu V Contemporary Mathematics and Its Applications. In Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic and Variational Methods. Hindawi Publ. Corp, New York; 2008.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.