Existence Of Solutions For A Fractional Laplacian Equation With Critical Nonlinearity

In this paper we study the existence of solutions of fractional Laplacian equation with critical exponent2*(s)=2N/(N-2s), N>2s and s∈(0,1). The functions f(x,u) and the parameter λ satisfy some assumptions respectively. We consider the following two cases.The first case:when f(x, u)=μu,μ∈R, λ>0, that is A(x) satisfies the following assumptions(Al) A∈C(RN,R), A≥0, and Ω:=int A-1(0) is a nonempty bounded set with smooth boundary, and Ω=A-1(0).(A2) There exists M0>0such that where L denotes the Lebesgue measure in RN.In (2), s∈(0,1) is fixed and (-△)s is the fractional Laplace operator, which may be defined asBy employing the variational method we prove the existence of nontrivial solu-tions for μ small and λ large. Every sequence of solutions of (2) concentrates at a solution of equationThe second case:By employing the variational method we prove the existence of nontrivial solutions for the equation when f(x, u) has subcritical growth with respect to u and λ=1in (1).This thesis consist of three chapters. The first chapter is devoted to discuss the introduction including research background and prerequisite knowledge. The second chapter deals with the existence of the solutions for the equation under the first case, the main conclusion is Theorem2.1.1and Theorem2.1.2. In the last chapter we research the the existence of the solutions for the equation about the second case, the main results are Theorem3.1.1.