The Research On The Existence Of Nontrivial Solutions To Nonlinear Elliptic Equations And Systems

In this paper, we mainly study the existence of nontrivial solutions to nonlin-ear Schrodinger-Kirchhoff equations with nonlinearities of subcritical and critical growth, the existence of multiple solutions to p-Laplacian type equations and the existence result for a semilinear system with zero mass and a coupled Schrodinger system with doubly critical exponents.The thesis consists of six 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 positive ground state solutions to the following nonlinear Kirchhoff problem: where a,b>0are constants and2<p<5. Under certain assumptions on V(x), we prove that (E1) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. Our main result especially solves problem (E1) in the case where p∈(2,3], which has been an open problem for Kirchhoff equation and can be viewed as a partial extension of a recent result of He and Zou in [62] concerning the existence of positive solutions to the Kirchhoff problem with a nonlinearity f(u)～|u|p-1u,3<p<5. We give a unified treatment for the proof of the existence result to (E1) with p∈(2,5).In Chapter Three, we consider the existence of positive solutions to the following nonlinear Kirchhoff type problem with critical Sobolev exponent: where a, b>0arc two constants. We assumed that f(x, t) satisfies two kinds of conditions:f(x,t)=f(t) is subcritical, superlinear at the origin and satisfies (AR) condition and f(t)/t3is strictly increasing or f(x,t)=fλ(x)|t|p-2t, fλ(x) is a sign-changing weight function and p∈[2,4). By using the mountain-pass theorem and the concentration compactness principle of P.L. Lions, we prove that (E2) has at least one positive solution. Our result partially extends a main result of [62] concerning Kirchhoff equations with subcritical nonlinearities.In Chapter Four, we study the existence of infinitely many solutions to the following quasilinear equation of p-Laplacian type in RN: with sign-changing radially symmetric potential V(x), where1<p<N, λ∈R and△pu=div(|Du|p-2Du) is the p-Laplacian operator, g(x, u)∈C(RN×R,R) is subcritical and p-superlinear at0as well as at infinity. We prove that under certain assumptions on V and g, problem (E3) has infinitely many solutions for any λ∈R by using a fountain theorem over cones under Ccrami condition. Our result can be seen as an extension of [49] concerning the existence of nontrivial solutions for the p-Laplacian problem on a bounded domain.In Chapter Five, we prove the existence of at least one positive solution pair (u,v)∈D1,2(RN) x D1-2(RN) to the following semilincar elliptic system-△u=K(x)f(v),x∈RN (S1)-△u=K(x)g(v),x∈RN by using a linking theorem, where K(x) is a positive function in LS(RN) for some s>1and the nonncgativc functions f,g∈C(R, R) arc of quasicritical growth, superlinear at infinity. We do not assume that f or g satisfies the (AR) condition as usual. Our main result provides a general existence result for the semilincar elliptic system with zero mass and can be viewed as a partial extension of a main result in [5] which deals with a single equation with zero mass case. Our result also extends the main result in [74] considering a semilincar elliptic system of Hamiltonian type in RN to zero mass case.In Chapter Six, we study the following critically coupled perturbed Brczis- Nirenberg problem where Ω (?) R3is a smooth bounded domain, λ1,λ2<0,μ1,μ2>0and β>0. Under certain conditions on λ1,λ2, there exists β1>0such that for any β>β1,(S2) has at least one positive least energy solution. In particular, we prove that (52) has positive least energy solutions of the form(c1ω, C2ω), where w is the least energy solution of the associated Brezis-Nirenberg problem. Our main result can be viewed as an extension of [41,42] considering the critically coupled perturbed Brezis-Nirenberg problem on bounded domains of RN (N≥4).