Multiple Solutions For Several Classes Of Nonlinear Differential Equations

This thesis studies the multiplicity of solutions for several class of nonlinear differen-tial equations in the whole space. The thesis contains six chapters.In Chapter1, we introduce the backgrounds and state the main results.In Chapter2, by using modified function technique and variational methods, we establish the existence of infinitely many homoclinic solutions for a second order Hamil-tonian system u-L(t)u+Fu(t,u)=0,(?)t∈R, where no coercive condition for F(t, u) at infinity is imposed.In Chapter3, we consider the following nonlinear sublinear Schrodinger equation at resonance: By using bounded domain approximation technique, we prove that the problem has in-finitely many solutions. We should point out that, we use a new method to prove the non-triviality and multiplicity of the solutions.In Chapter4, we consider the problem where λ>0,4<p<22*,2*=2N/N-2,N≥3. By using bounded domain approximation technique, we prove that the problem has infinitely many solutions. As a main novelty with respect to some previous results, we do not require any periodicity or symmetry conditions on the potential V(x).In Chapter5, we prove the existence of radial solutions with arbitrarily many sign changes for quasilinear Schrodinger equation: where N≥3, p∈(1,3N+2/N-2).The proof is accomplished by minimization under a conve-nient constraint. In Chapter6, we state the conclusions and the prospects of this thesis.