A compact embedding result and its applications to a nonlocal Schrödinger equation

Abstract

AbstractCompactness is a fundamental issue in nonlinear analysis. Assume and with , we consider a subspace of , which is the space of invariant functions corresponding to a subgroup G of , where is a kind of function spaces including fractional Sobolev spaces . We show that the embeddings are compact for provided Ω is compatible with G, where is the fractional Sobolev critical exponent. Moreover, the existence and multiplicity of radial and nonradial solutions were obtained of the following nonlocal Schrödinger equation: where is an integro‐differential operator has order 2s and can be given by and the kernel K satisfies the following properties: there is and such that for any ; , where .