Existence results for zero mass polyharmonic system

Abstract

ABSTRACT In this paper, we deal with the following polyharmonic system in the zero-mass case where K is a positive weight function and f, g are two continuous functions with critical or supercritical growth at zero and satisfying a quasicritical growth at infinity. We give a global condition which is weaker than the Ambrosetti–Rabinowitz condition and we point out its importance for checking the Palais Smale compactness condition. By applying a corollary of Theorem 2.1 in [Li GB, Szulkin A. An asymptically periodic Schrödinger equation with indefinite linear part. Commun Contemp Math. 2002;4:763–776] for strongly indefinite functionals, we prove the existence of at least one nontrivial pair solution under weaker hypotheses on the nonlinear terms. The present paper extend previous results of He [Nonlinear Schrödinger equations with sign-changing potential. Adv Nonlinear Stud. 2012;12:237–253] and Li-Ye [Li GB, Ye H. Existence of positive solutions to semilinear elliptic systems in RN with zero mass. Acta Math Sci. 2013;33(4):913–928].