Infinitely many solutions for a class of quasilinear equation with a combination of convex and concave terms

Kaimin Teng, Ravi P. Agarwal

DOI: http://dx.doi.org/10.12775/TMNA.2017.031

Abstract


We consider the following quasilinear elliptic equation with convex and concave nonlinearities: \begin{equation*} -\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N, \end{equation*} where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and $g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function. Under some suitable conditions on $V,K$ and $g$, the existence of infinitely many solutions is established.

Keywords


Quasilinear Schrödinger equation; Palais-Smale condition; concave-convex nonlinearities

Full Text:

PREVIEW FULL TEXT

References


M.J. Alves, P.C. Carrião and O.H. Miyagaki, Non-autonomous perturbation for a class of quasilinear elliptic equations on R, J. Math. Anal. Appl. 344 (2008), 186–203.

C.O. Alves and G.M. Figueiredo, Multiple solutions for a quasilinear Schrödinger equation on RN , Acta Appl. Math. 136 (2015), 91–117.

C.O. Alves, G.M. Figueiredo and U.B. Severo, Multiplicity of positive solutions for a class of quasilinear problems, Adv. Differential Equations 14 (2009), 911–942.

A. Ambrosetti, J. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219–242.

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543.

A. Ambrosetti and Z.Q. Wang, Positive solutions to a class of quasilinear elliptic equations on R, Discrete Contin. Dyn. Syst. 9 (2003), 55–68.

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc. 123 (1995), 3555–3561.

A.V. Borovskiı̆ and A.L. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP 77 (1993), 562–573.

P.C. Carriao, R. Lehrer and O.H. Miyagaki, Existence of solutions to a class of asymptotically linear Schrödinger equations in RN via the Pohozaev manifold, J. Math. Anal. Appl. 428 (2015), 165–183.

S.X. Chen, Existence of positive solutions for a class of quasilinear Schrödinger equations on RN , J. Math. Anal. Appl. 405 (2013), 595–607.

C.D. Clark, A variant of the Lusternik–Schnirelman theory, Indiana Univ. Math. J. 22 (1972), 65–74.

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal. 56 (2004), 213–226.

J.M. Do Ó and U.B. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal. 8 (2009), 621–644.

Y.X. Guo and Z.W. Tang, Ground state solutions for the quasilinear Schrödinger equation, Nonlinear Anal. 175 (2012), 3235–3248.

A.M. Kosevich, B.A. Ivanov and A.S. Kovalev, Magnetic solitons, Physics Reports 194 (1990), 117–238.

S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan 50 (1981), 3262–3267.

A.G. Litvak and A.M. Sergeev, One dimensional collapse of plasma waves, JEPT Letters 27 (1978), 517–520.

J.Q. Liu and Z.Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc. 131 (2003), 441–448.

J.Q. Liu and Z.Q. Wang, Multiple solutions for quasilinear elliptic equations with a finite potential well, J. Differential Equations 257 (2014), 2874–2899.

J.Q. Liu, Y.Q. Wang and Z.Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations 187 (2003), 473–493.

J.Q. Liu, Y.Q. Wang and Z.Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), 879–901.

J.Q. Liu, Z.Q. Wang and Y.X. Guo, Multibump solutions for quasilinear Schrödinger equations, J. Func. Anal. 262 (2012), 4040–4102.

J.Q. Liu, Z.Q. Wang and X. Wu, Multibump solutions for quasilinear elliptic equations with critica growth, J. Math. Phys. 54 (2013), 121501.

S.B. Liu and S.J. Li, An elliptic equation with concave and convex nonlinearities, Nonlinear Anal. 53 (2003), 723–731.

X.Q. Liu, J.Q. Liu and Z.Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013), 253–263.

M. Poppenberg, K. Schmitt and Z.Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 329–344.

P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math.. vol. 65, Amer. Math. Soc., Providence, 1986.

M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.

U.B. Severo, Multiplicity of solutions for a class of quasilinear elliptic equations with concave and convex terms in R, Electron. J. Qual. Theory Differential Equations 5 (2008), 1–16.

U.B. Severo, Existence of weak solutions for quasilinear elliptic equations involving the pLaplacian, Electron. J. Differential Equations 56 (2008), 1–16.

M.Z. Sun, J.B. Su and L.G. Zhao, Infinitely many solutions for a Schrödinger–Poisson system with concave and convex nonlinearities, Discrete Contin. Dynam. Systems 35 (2015), 427–440.

S. Takeno and S. Homma, Classical planar heisenberg ferromagnet, Complex Scalar Fields and Nonlinear excitation, Prog. Theoret. Phys. 65 (1981), 172–189.

E. Tonkes, A semilinear elliptic equation with convex andconcave nonlinearities, Topol. Methods Nonlinear Anal. 13 (1999), 251–271.

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.

C.X. Wu and T.F. Wang, Orlicz Spaces and Applications, Heilongjiang Science and Technology Press, 1983.

X. Wu and K. Wu, Infinitely many small energy solutions for a modified Kirchhoff-type equation in RN , Comput. Math. Appl. 70 (2015), 592–602.

L.R. Xia, M.B. Yang and F.K. Zhao, Infinitely many solutions to quasilinear elliptic equation with concave and convex terms, Topol. Methods Nonlinear Anal. 44 (2014), 539–553.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism