Solutions To Several Kinds Of Elliptic Equations With Non-local Terms

In this paper, we investigate the existence and multiplicity of solutions for several kinds of elliptic partial differential equations with non-local terms by variational methods.We firstly investigate the existence and multiplicity of solutions of Schrodinger-Poisson system where the potential k(x) allows sign changing. By Nehari manifold method and concentration-compactness principle,under different methods and conditions from, we obtain the existence and multiplicity of solutions for the system.Next, we study the existence and multiplicity of solutions of the quasilinear Schrodinger equation where the potential V allows sign changing and the nonlinearity satisfies conditions weaker than the classical Ambrosetti-Rabinowitz (abbreviated as AR) condition. By a local linking theorem and the fountain theorem, we obtain the existence and multiplicity of solutions for the equation.Third, we investigate solutions for the field equation involving a fractional Laplacian By mountain pass theorem jointly with a rearrangement argument, we prove existence of groud state solution without the classical AR condition.Finally, we investigate the existence and multiplicity of solutions to Kirchhoff equation where the potential k(x) allows sign changing. Making use of Nehari manifold method and concentration compactness principle, we obtain the existence and multiplicity of solutions for this equation. The case can not be dealt with by the arguments of. But our main results can be viewed as partial extensions of theirs.