Articles

EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS

  • Quanqing LI ,
  • Xian WU
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  • 1. Department of Mathematics, Honghe University, Mengzi 661100, China;
    2. Department of Mathematics, Yunnan Normal University, Kunming 650092, China

Received date: 2016-02-29

  Revised date: 2017-04-29

  Online published: 2017-12-25

Supported by

This work was supported in part by the National Natural Science Foundation of China (11501403; 11461023) and the Shanxi Province Science Foundation for Youths under grant 2013021001-3.

Abstract

In this paper, we study the following generalized quasilinear Schrödinger equations with critical or supercritical growths
-div(g2(u)∇u) + g(u)g'(u)|∇u|2 + V (x)u=f(x, u) + λ|u|p-2u, x ∈ RN,
where λ > 0, N ≥ 3, g:R → R+ is a C1 even function, g(0)=1, g'(s) ≥ 0 for all s ≥ 0,???20170623???((g(s))/(|s|α-1)):=β > 0 for some α ≥ 1 and (α-1)g(s) > g'(s)s for all s > 0 and pα2*. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 using a change of variables and variational method.

Cite this article

Quanqing LI , Xian WU . EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS[J]. Acta mathematica scientia, Series B, 2017 , 37(6) : 1870 -1880 . DOI: 10.1016/S0252-9602(17)30113-3

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