Existence And Multiplicity Of Solutions To Two Dierential Equations With Convex And Concave

In this thesis, by using some approaches in variational method, such as moun-tain pass theorem, dual mountain pass theorem, fountain theorem, dual fountain theorem, we study the existence and multiplicity of solutions to two differential equations with convex and concave.First, we consider a class of impulsive equation with concave and convex bound-ary value problemswhere0=t0<t1<t2<…<tp<tp+1=T,â–³,u’(tj))=u’(t+j)-u’(t-j)=lim sâ†'t+j u’(s)-lim sâ†'t-j u’(s).By using mountain pass theorem and symmetric mountain pass theorem sepa-rately, we obtain that the problem (3.1.1) has at least one solution and infinitely many solutions.Second, we consider the existence and multiplicity of solutions to a class of Schrodinger equation with concave and convexwhere1<p<2<q<2*=2N/N-2,,N≥3,u∈R\{0}.By using fountain theorem and dual fountain theorem separately, we obtain that the problem (4.1.1) has infinitely many solutions.This thesis contains four chapters.In chapter one, we review the purpose and significance, background and previ-ous results, the main problems we considered.In chapter two, we recall some basic knowledge about variational method, and some basic lemmas which we will use in the following chapters. In chapter three, we discuss the existence and multiplicity of solutions to a class of impulsive equation with concave and convex, using mountain pass theorem and symmetric mountain pass theorem.In chapter four, we discuss the existence and multiplicity of solutions to a class of Schrodinger equation with concave and convex, using fountain theorem, dual fountain theorem.