Existence and Multiplicity of Solutions for a Bi-Non-Local Problem

The aim of this paper is to investigate the existence and multiplicity of solutions for a binon-local problem. Precisely, we show that the above problem admits at least a non-trivial positive energy solution by using the mountain pass theorem. Furthermore, with the help of the fountain theorem, we obtain the existence of infinitely many positive energy solutions, assuming a symmetric condition for g. The main feature and difficulty of this paper is the presence of a double non-local term involving two variable parameters.