Positive solutions for P-Laplace problems with nonlinear time-fractional differential equation

In this paper, we study the existence and multiplicity of positive solutions for semi-linear elliptic equations with a sign-changing weight function in weighted Sobolev spaces. By investigating the compact embedding theorem and based on the extraction of the Palais-Smale sequence in the Nehari manifold which is a subset of the weighted Sobolev spaces, we derive the existence of the multiple positive solutions of the equations by using the variational method. In the last part of this paper, by applying the Arzela-Ascoli fixed point theorem, some existence results of the corresponding time-fractional equations for semi-linear elliptic equations are obtained.