The Existence Of Infinitely Many Solutions For Two Elliptic Partial Differential Equations

In this thesis, we study the existence of infinitely many solutions for two elliptic partial differential equations by applying variational methods including descending flow invariant sets, minimax method and symmetric mountain pass theorem.In the introduction, we review some background and known results about the problems.In chapter one, we introduce some basic knowledge and lemmas of critical point theory and give some notations.In chapter two, we consider the existence of nodal solution for an semilinear elliptic boundary value problem whereλ≥0 is a parameter,Ωis a bounded domain in RN with smooth boundary aΩ. The nonlinearity involves a combination of concave and convex terms. We construct certain invariant sets of the gradient flow so that all positive and negative solutions are contained in these invariant sets and that minimax procedure can be used to construct nodal critical point of the energy functional outside of these invariant sets.In chapter three, we use the symmetric mountain pass theorem to obtain the existence of infinitely many solutions for the semilinear Schrodinger equation The potential V(x) is sign-changing.In chapter four, we summarize the main results of this thesis.