Research On The Existence Of Solutions For Some Elliptic Equations And Systems

In this paper, we mainly study the existence of solutions for some semilinear elliptic equations and systems, some nonlinear Schrodinger equations with electro-magnetic fields and nonlinear Schrodinger-Poisson system.The thesis consists of seven chapters:In Chapter One, we summarize the background of the related problems and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.In Chapter Two, we study the existence of a nontrivial solution to the following nonlinear elliptic problem: whereΩis a bounded domain of RN and a∈LN/2(Ω),N≥3,f∈C0(CΩ×(Q×R1,R1) is superlinear at t=0 and subcritical at t=∞. Under suitable conditions, the equation (S1) possesses the so-called linking geometric structure. We prove that the equation (S1) has at least one nontrivial solution without assuming the Ambrosetti-Rabinowitz condition. Our main result extends a recent result of Miyagaki and Souto given in [103] for the equation (S1) with a(x)=0 and possessing the mountain-pass geometric structure.In Chapter Three, we prove the existence of at least one positive solution pair (u,v) E Hl(RN)×(RN) to the following semilinear elliptic systemby using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functionsf, g∈C0(RN×R1) are that, f(x,t) and g(x,t) are superlinear at t=0 as well as at t=+∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. We generalize the results in [103] from a single equation to the system. In Chapter Four, we study the following semilinear elliptic system (S2) further. Here f(x,t) and g(x,t) satisfy some different conditions from Chapter Three. By using critical point theory of strongly indefinite functionals, we obtain a positive ground state solution for (S2). Moreover, if f(x,t) and g(x,t) are periodic in x and odd in t, then (S2) has infinitely many geometrically distinct solutions. We generalize the results in [88,89].In Chapter Five, we study the nonlinear Schrodinger equation with electromag-netic fieldswhere the vector A(r)=(A1(r), A2(r),…, AN(r)) is such that Aj(r)(j=1,2,…, N) is a real function on R+ and V(r) is a positive function on R+,1＜p＜N+2/N-2 if N≥3 and 1＜p＜+∞ifN=1,2. We prove that the equation (E1) has infinitely many non-radial complex-valued solutions under conditions (H1) and (H2). Our main re-sult extends a result of Wei and Yan given in [134] for the equation (S1) with A(y)= 0.In Chapter Six, we are concerned with the existence of multi-bump solutions for a nonlinear Schrodinger equations with electromagnetic fields where 2＜p＜2N/N-2 if N≥3 and 2＜p＜+∞if N= 1,2 and∈＞0 is a parameter. a(x) is a positive continuous function on RN, and A(x) = (A1(x), A2(x),..., AN(x)) is such that Aj(x)(j= 1,2,..., N) is a real function on RN. We prove under some suitable conditions that for any positive integer m, there exists∈(m)> 0 such that, for 0＜∈＜∈(m), the problem (E2) has an m-bump complex-valued solution. As a result, when∈→0, the equation has more and more multi-bump complex-valued solutions. Our main result extends a result of Liu and Lin given in [96] for the equation (S2) with A(x)≡0.In Chapter Seven, we study the following nonlinear Schrodinger-Poisson system where K(x) is positive and continuous function in R3, and lim K(x)=0,2＜p＜6 and (?)＞0 is a parameter. For any positive integer m, we prove that there exists (?)(m)＞0 such that, for 0＜(?)＜(?)(m), equation (SP) has an m-bump positive solution under some suitable conditions. As a consequence, equation (SP) has more and more multi-bump positive solutions as (?)→0. We generalize the result in [96] from a single nonlinear Schrodinger equation to the nonlinear Schrodinger-Poisson system.