Infinitely many solutions for a class of fractional equations via variant fountain theorems

Abstract

This article is concerned with a class of fractional type equation involving an anisotropic operator and potential of the form $$(-\Delta_x)^s u-\Delta_yu+\Phi(x,y)u=g(x,y,u),\;(x,y)\in\mathbb{R}^n\times\mathbb{R}^m.$$ By means of the variational method and the variant fountain theorems, we investigate the existence of infinitely many high or small energy solutions without the usual assumption of coerciveness on the potential $\Phi$ in the different cases when the nonlinear term is either asymptotically linear or superquadratic growth.