Infinitely Many Solutions for the Fractional <i>p</i>&<i>q</i>-Laplacian Problems in <i>R</i>^N

Abstract

In this paper, we consider the following class of the fractional p&q-Laplacian problem: (−Δ)psu+(−Δ)qsu+V(x)(|u|p−2u+|u|q−2u)+g(x)|u|r−2u=K(x)f(x,u)+h(u),x∈RN,V:RN→R+ is a potential function, and h:R→R is a perturbation term. We studied two cases: if f(x,u) is sublinear, by means of Clark’s theorem, which considers the symmetric condition about the functional, we get infinitely many solutions; if f(x,u) is superlinear, using the symmetric mountain-pass theorem, infinitely many solutions can be obtained.

Keywords

• fractional <i>p</i>&<i>q</i>-Laplacian
• without the (AR) condition
• infinitely many solutions