## Topological Methods in Nonlinear Analysis

### Infinitely many solutions to quasilinear elliptic equation with concave and convex terms

#### Abstract

In this paper, we are concerned with the following quasilinear elliptic equation with concave and convex terms $$-\Delta u-{\frac12}\,u\Delta(|u|^2)=\alpha|u|^{p-2}u+\beta|u|^{q-2}u,\quad x\in \Omega, \tag(\rm P)$$ where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $1< p< 2$, $4< q\leq 22^*$. The existence of infinitely many solutions is obtained by the perturbation methods.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 44, Number 2 (2014), 539-553.

Dates
First available in Project Euclid: 11 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1460381369

Mathematical Reviews number (MathSciNet)
MR3328355

Zentralblatt MATH identifier
1365.35042

#### Citation

Xia, Leran; Yang, Minbo; Zhao, Fukun. Infinitely many solutions to quasilinear elliptic equation with concave and convex terms. Topol. Methods Nonlinear Anal. 44 (2014), no. 2, 539--553. https://projecteuclid.org/euclid.tmna/1460381369

#### References

• 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, 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), 55–66.
• T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), 1205–1216.
• T. Bartsch and M. Willem, Periodic solutions of nonautonomous Hamiltonian systems with symmetries, J. Reine Angew. Math. 451 (1994), 149–159.
• ––––, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc. 123 (1995), 3555–3561.
• H. Bellout, On a special Schauder basis for the Sobolev spaces $W_0^{1,p}(\Omega)$, Illinois J. Math. 39 (1995), 187–195.
• H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
• H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
• A. Canino and M. Degiovanni, Nonsmooth critical point theory and quaslinear Schrödinger elliptic equations, Topol. Methods in Differential Equations and Equations and Inclusions (Montreal, PQ, 1994), 472 (1995), 1–50.
• M. Colin and L. Jeanjean, Solutions for quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. 56 (2002), 392–344.
• J.M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations 248 (2010), 722–744.
• J.M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal. 8 (2009), 621–644.
• ––––, Solitary wave for a class of quasilinear quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differential Equations 69 (2010), 397–408.
• M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler, Banach Space Theory. The Basis for Linear and Nonlinear Analysis. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011.
• S. Fučík, O. John and J. Nečas, On the existence of Schauder bases in Sobolev spaces, Comment. Math. Univ. Carolinae 13 (1972), 163–175.
• A. J. García and A.I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 877–895.
• H.F. Lins and E.A.B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal. 71 (2009), 2890–2905.
• J. Liu and Z.-Q. Wang, Solition solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc. 131 (2003), 329–344.
• J. Liu, Y. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), 879–901.
• ––––, Solition solutions for quasilinear Schrödinger equations II, J. Differential Equations 187 (2003), 473–493.
• X. Liu, J. Liu and Z.-Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc. 141 (2013), 253–263.
• A. Moameni, Existence of solition solutions for a quasilinear Schrödinger equations involing critical growth in $\mathbb{R}^N$, J. Differential Equations 229 (2006), 570–587.
• M. Poppenburg, K.Schmitt and Z.-Q. Wang, On the existence of solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 392–344.
• E.A.B. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Nonlinear Anal. 72 (2010), 2935–2949.
• M. Willem, Minimax Theorem, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.