Existence and multiplicity results for double phase problem on compact Riemannian manifolds

Abstract

In this paper, we consider the following double phase problem:Δp,gu+a(x)Δq,gu+|u|p−2u+a(x)|u|q−2u=f(x,u),x∈M, where M is a compact Riemannian manifold without boundary with dimension N≥3 and 1<p<q<N. We investigate the properties of double phase operator on M and obtain the existence and multiplicity results under the different assumptions on f(x,u) with the help of the variational approach.