On The Solutions For Some Nonlinear Elliptic Equations (systems)

In this thesis, we mainly deal with the problems on the existence of solutions for some nonlinear elliptic equations and systems. There are five chapters in this thesis.In Chapter one, we summarize the background of the related problems, and state the main results of the present thesis.In Chapter two, we consider the following nonlinear Schrodinger system with mixed couplings in R3 where λi,μi>0,βij=βji(i,j=1,…,N,i≠J).The system appears in modeling of Bose-Einstein condensates theory. While most existing works in the literature concern with purely attractive or purely repulsive couplings (i.e, all βij have the same signs), we examine the effect of mixed nonlinear couplings on the solution structure and obtain vector solutions with some of the components synchronized between them while being segregated with the rest of the components simultaneously.The Chapter three is concerned with the following linearly coupled system where ε>0 is a small parameter, Pi(x) are positive potentials and λij=λji> 0 (i≠j) are coupling constants. We investigate the effect of potentials and the linear coupling constants to the solution structure. Moreover we prove that for any positive integer k∈Z, we construct k spikes solution concentrating near the local maximum point x0i of Pi(x). When x0i=x0j, Pi(x0i)=Pj(x0j)=a, i≠j, i, j=1,…, N, the components have spikes clustering at the same point as εâ†'0+. When x0i≠x0j,i≠j, the components have spikes clustering at the different points as εâ†'0+.In Chapter four, we consider the following nonlinear fractional Schrodinger equation ε2s(-Δ)su+V(x)u=up,u>0 in RN, where ε> 0 is a small parameter, V(x) is a positive function,0<s<1,1<p< N+2s/N-2s. For any positive integer k ∈ Z+, we can construct k spikes solution near the local maximum point of V(x).In Chapter five, we study the following elliptic system with critical exponent where Ω is a smooth bounded domain in RN, N=5,2*:=2N/N-2 is the critical Sobolev exponent,μ1,μ2>0,β∈(-(?),0),0<λ1,λ2<λ1(Ω),λ1(Ω) is the first eigenvalue of -Δ in H01(Ω). In [32], Chen, Lin and Zou established a sign-changing solution for the above system in the case N≥6 for β<0 and λ1,λ2 ∈ (0,λ1(Ω)). We show that in dimension N=5, for λ1 andλA2 slightly smaller than λ1(Ω), the above system has a sign-changing solution in the following sense:one component changes sign and has exactly two nodal domains, while the other one is positive.