Research On The Existence Of Solutions For Three Kinds Of Elliptic Partial Differential Equations

In recent years,the research on the Kirchhoff-type equation and the Schrodinger equation has been an important topic for many scholars,such as the existence of infinitely many solutions,positive solutions,and other properties of solutions.The results are widely used in the fields of fluid mechanics,nonlinear optics and elastic mechanics.Therefore,the research on elliptic partial differential equations has very important theoretical value.This thesis mainly studies three kinds of elliptic partial differential equations via variational methods.Under the suitable assumptions of nonlinearities and potentials,the existence and multiplicity of solutions are proved,which extend and improve the results in related literatures.This thesis is divided into the following four chapters:In chapter one,we introduce the research background,the present status of this thesis,and expound the research work,the related preparatory knowledge in this thesis.In chapter two,we study the Kirchhoff-type equation with general potentials-(a+b?R3|?u|2dx)?u+V(x)u=K(x)| u |p-2 u+f(x,u),x ? R3,where constants a,b>0,p?(4,6),V,K:R3?R are two potentials.Under general superlinear assumptions on nonlinearity f(x,u),we establish the existence of infinitely many solutions via Symmetric Mountain Pass Theorem.The third chapter mainly studies the degenerate quasilinear Schrodinger equation with general potentials-div(a(x,?u))+V(x)| x|-?p*|up-2 u=K(x)| x|-?p*f(x,u),x?RN,where N?3,1<p<N,-?<?(N-p)/p p,??e??+1,d-1+?-e,p*:=p*(?,e)-Np/(N-dp)(critical Hardy-Sobolev exponent),V and K are nonnegative potentials.The function f satisfies weaker assumptions than the(AR)condition.When the function a satisfies suitable assumptions,the existence of infinitely many nontrivial solutions for the equation is proved by using variational methods.In chapter four,we study a class of generalized quasilinear Schrodinger equation involving concave and convex nonlinearities-div(g2(u)?u)+g(u)g'(u)| ?u |2+V(x)u-?f(x,u)+h(x,u),x?RN,where ?>0,N?3,2*=2N/(N-2),g?C1(R,R+).By using a change of variable,we obtain the existence of two different positive solutions for this class of equation via the Mountain Pass Theorem.