Research On The Existence, Concentration And Multiplicity Of Nontrivial Solutions For Several Classes Of Singularly Perturbed Elliptic Equations

In this paper, we mainly study the existence, concentration and multiplicity of nontrivial solutions to several classes of singularly perturbed Kirchhoff type equa-tions, Schrodinger-Poisson equations and quasilinear Schrodinger equations.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 and Chapter Three, we study two classes of singularly per-turbed Kirchhoff type equations.In Chapter Two, we study the existence, concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth where ε is a small positive parameter and a, b> 0 are constants,f∈C1(R+,R) is subcritical, V:R3→R is a locally Holder continuous function. We first prove that for ε>0 sufficiently small, the above problem has a weak solution uε with exponential decay at infinity by using penalization method due to [M. del Pino, P. L. Felmer, Calc. Var. Partial Differential Equations 4 (1996) 121-137]. Moreover, uε concentrates around a local minimum point of V as ε→0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(x) attains its local minima.We point out that to overcome the obstacle due to the appearance of the critical nonlinearity u5, we need to pull the energy level down below a certain critical level. In [J. Wang, L. Tian, J. Xu, F. Zhang, J. Differential Equations 253 (2012) 2314-2351], J. Wang el al. give a critical level where S is the best Sobolev constant of the imbedding D1,2(R3)→L6(R3). But the energy level c* is not the best. In this chapter, we prove that the precise threshold value for the Kirchhoff type equation is: Our result can be viewed as a partial extension of [X. He, W. Zou, J. Differential Equations 2 (2012) 1813-1834] concerning Kirchhoff type equations with subcritical nonlinearities.In Chapter Three, we are concerned with the following Kirchhoff type equation with critical nonlinearity: where ε is a small positive parameter, a, b>0, λ>0,2<p≤4. Under certain assumptions on the potential V, we construct a family of positive solutions uε∈ H1(R3) which concentrates around a local minimum of V as ε→0.Critical growth Kirchhoff type problem has been studied in [Y. He, G. Li, S. Peng, Adv. Nonlinear Stud.14 (2014), 441-468], where the assumption for f(u) is that f(u)～|u|p-2u with 4<p<6 and satisfies the Ambrosetti-Rabinowitz condition ((AR)condition in short) which forces the boundedness of any Palais-Smale sequence ((PS) condition in short) of the corresponding energy functional of the equation. As g(u):=λ|u|p-2u+|u|4u with 2<p≤4 does not satisfy the (AR) condition ((?)μ>4,0<μ∫0ug(s)ds≤g(u)u), the boundedness of (PS) sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function g(s)/s3 is not increasing for s>0 prevents us from using the Nehari manifold directly as usual. Our result extends the main result in [Y. He, G. Li, S. Peng, Adv. Nonlinear Stud.14 (2014),441-468] concerning the existence and concentration of positive solutions to the case where f(u)～|u|p-2u with 4<p<6.In Chapter Four, we are concerned with the following Schrodinger-Poisson e-quation with critical nonlinearity: where ε> 0 is a small positive parameter, λ>0,3<p≤4. Under certain assumptions on the potential V, we construct a family of positive solutions uε∈ H1(R3) which concentrates around a local minimum of V as ε→0.Subcritical growth Schrodinger-Poisson equation has been studied extensively, where the assumption for f(u) is that f(u)～|u|p-2u with 4<p<6 and satisfies the (AR) condition which forces the boundedness of any (PS) sequence of the corresponding energy functional of the equation. The more difficult critical case is studied in this paper. As g(u):=λ|u|p-2u+|u|4u with 3<p≤4 does not satisfy the (AR) condition ((?)μ>4,0<μ∫0ug(s)ds≤g(u)u), the boundedness of (PS) sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function s3/g(s) is not increasing for s>0 prevents us from using the Nehari manifold directly as usual. The main result we obtained in this paper is new.In Chapter Five, we study the existence, concentration and multiplicity of weak solutions to the quasilinear Schrodinger equation with critical Sobolev growthwhere ε is a small positive parameter, N≥3,2*=N-2/2N,4<q<2·2*, min V> 0 and inf W>0. Under proper assumptions, we obtain the existence and concentration phenomena of soliton solutions of the above problem. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple soliton solutions by employing the topology of the set where the potentials V(x) attains its minima and W(x) attains its maxima.Our result can be seen as a partial extension of [X. He, A. Qian, W. Zou, Non-linearity 26 (2013),3137-3168] concerning the following type singularly perturbed quasilinear Schrodinger equationwhere h(u) is of subcritical growth and the potential V(x) satisfies the following condition due to the celebrated work [P. Rabinowitz, Z. Angew. Math. Phys.43 (1992) 270-291]:...