On The Existence And Multiplicity Of Solutions For Several Class Of Semilinear Differential Equations

This thesis mainly studies the existence and multiplicity results for three class of semilinear differential equations in the whole space.This thesis consists of four chapters.In Chapter1,we summarize the related research background and state our main results.In Chapter2,we consider the existence of homoclinic solutions for second-order Hamiltonian system ü-L(t)u+Wu(t, u)=0,(?)t∈R. Assuming that L(t)possesses certain compactness condition, under a variant of Ahmad-Lazer-Paul condition, we obtain the existence of infinitely many small energy homoclinic solutions by using the fountain theorem dual version.For the super-quadratic case,under a variant of monotonicity condition, we establish the existence of infinitely many high energy homoclinic solutions by using a variant of fountain theorem.Moreover,we discuss the periodic and asymptotically periodic cases.In Chapter3,we investigate the existence of homoclinic solutions for first-order Hamiltonian system u=(?)Hu(t,u),(?)t∈R. Suppose that H(t,u) is periodic in t. Under a new class of super-quadratic condition,we prove the existence of the least action homoclinic solution by using a generalized linking theorem. Moreover, H(t,u) is even in v,we obtain infinitely many geometrically distinct homoclinic solutions.We also discuss the existence and multiplicity results for a class of asymptotically quadratic first-order Hamiltonian system.In Chapter4,we study the existence of ground state solutions for the following Schrodinger equation where V(x) is periodic in x,the nonlinearity f(x,u) is non-periodic in x. Under a new super-quadratic condition, we consider the limit equation,and obtain the ground state solution and the least energy c.We show that the corresponding functional satisfies the (C)c-condition,if c<c. Consequently, we can prove the existence of ground state solution by a generalized linking theorem.