Multiple Solutions For Some Semilinear Elliptic Equations And Systems In R~n

In this thesis, by using some approach in variational methods such as minimax principle, Nehari manifold and concentration-compactness principle, we study mul-tiplicity of solutions to two semilinear elliptic equations and one system of semilinear elliptic equations in RN.First, we consider the multiplicity of solutions for a semilinear elliptic equation: where V(x) maybe change sign, by using symmetric mountain pass theorem, we obtain infinitely many solutions for equation (P1).Second, we consider the multiplicity of solutions for a nonhomogeneous semi-linear elliptic equation: by using mountain pass theorem and concentration-compactness principle, we obtain at least two positive solutions for equation (P2).Finally, we discuss a semilinear elliptic system: by using decomposition of Nehari manifold and concentration-compactness principle, we obtain at least two nontrivial nonnegative solutions for system (Eλ,μ).The thesis consists of five chapters.In chapter one, we review some background and results on above mentioned elliptic equations and system.In chapter two, we recall some basic knowledge of Sobolev spaces, some basic lemmas and give some notations.In chapter three, we discuss the existence of infinitely many solutions for a semilinear elliptic equation. In chapter four, we discuss multiple solutions for a nonhomogeneous semilinear elliptic equation without AR condition.In chapter five, we obtain multiplicity of solutions for a semilinear elliptic system by using the technique of Nehari manifold.