## Topological Methods in Nonlinear Analysis

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

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 50, Number 1 (2017), 299-332.

Dates
First available in Project Euclid: 14 October 2017

https://projecteuclid.org/euclid.tmna/1507946582

Digital Object Identifier
doi:10.12775/TMNA.2017.031

Mathematical Reviews number (MathSciNet)
MR3706163

Zentralblatt MATH identifier
06851002

#### Citation

Teng, Kaimin; Agarwal, Ravi P. Infinitely many solutions for a class of quasilinear equation with a combination of convex and concave terms. Topol. Methods Nonlinear Anal. 50 (2017), no. 1, 299--332. doi:10.12775/TMNA.2017.031. https://projecteuclid.org/euclid.tmna/1507946582

#### References

• M.J. Alves, P.C. Carrião and O.H. Miyagaki, Non-autonomous perturbation for a class of quasilinear elliptic equations on $\mathbb{R}$, J. Math. Anal. Appl. 344 (2008), 186–203.
• C.O. Alves and G.M. Figueiredo, Multiple solutions for a quasilinear Schrödinger equation on $\mathbb{R^{N}}$, 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 $\mathbb{R}$, 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. Borovskiĭ 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 $\mathbb{R}^N$ 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 $\mathbb{R}^N$, 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.
• ––––, 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.
• ––––, 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 $\mathbb{R}$, Electron. J. Qual. Theory Differential Equations 5 (2008), 1–16.
• ––––, Existence of weak solutions for quasilinear elliptic equations involving the $p$-Laplacian, 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 $\mathbb{R}^N$, 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.