Existence And Concentration Of Infinitely Many Sign-changing Solutions For Semiclassical Choquard Equation

In this thesis,by using the variational method,especially the critical point theory,we study the existence,multiplicity and concentration phenomenon of sign-changing solutions for semiclassical Choquard equation,and obtain some new results.The thesis consists of three chapters.In Chapter one,we mainly introduce the physical background and the current research progress of the Choquard equation,as well as the preliminary results and notations needed in the whole thesis.In Chapter two,we study the following semiclassical Choquard equation with subcritical growth where Δp is the p-Laplacian operator,Δpv=▽(|▽v|p-2▽v),N≥3,1<p<N,0<α<min {2p,N-1},p<q<pα*,pα*=(p(2N-α))/(2(N-p)),ε>0 is a small parameter,the potential function V is bounded.We prove the existence of infinitely many sign-changing solutions by using the method of invariant sets of descending flow and the perturbation method,and establish that these solutions concentrating near the critical points of the potential function V as ε→0 by using the penalization method.In Chapter three,we consider the following semiclassical Choquard equation with critical growth where>0,N≥3,0<α<min{4,N-1},max{2,2*-1}<q<2*,2α*=(2N-α)/(N-2),ε>0 is a small parameter,and the potential function V is bounded.We prove the existence of infinitely many sign-changing solutions by using the method of invariant sets of descending flow and the truncation technique,and establish these solutions concentrating near the critical points of the potential function V as ε→0 by using the penalization method.