A Class Of Nonlinear Elliptic Equations With Non-homogeneous Boundary Conditions

In this paper, we consider the existence and multiple solutions of the following nonlinear elliptic equation with non-homogeneous boundary conditions:where RN (N 2) is an open bounded domain with smooth boundary, f(x) L2( ) and g(x) H1/2( ) are given functions, , are two real parameters, p, q are real numbers which are greater than 1, moreover, p q.It is quite evident that the ranges of values which parameters , as well as real numbers p, q influence on studying Equation (P9), especially, the multiplicity of solutions of Equation (Pg).If = 1, Equation (Pg) is changed intoWhen 0, 2 q < p < 2(N + 1)/N, Candela-Salvatore [11] give multiple solutions of Equation (A1.1) via perturbative methods which introduced by Bahri-Berestycki [7], Rabinowitz [8] and Struwe [9] in studying of the following equation with Dirichlet boundary conditions:Inspired by [11], in this paper, we study the existence and multiple solutions of Equation (Pg) by using variational methods and perturbative methods for the various cases that , R and 1 < p, q < 2N/(N - 2).Our results can be included in the following:Theorem 1.1 Let f L2( ), g H1/2( ) be given, 0 and , p, q be such thatThen Equation (Pg) has infinitely many solutions, if p, q satisfy thatThen Equation (Pg) has at least one solution.Corollary 1.1 Let f, g be as in Theorem 1.1, 0 and A, p, q be such that Then Equation (Pg) has infinitely many solutions.Theorem 1.2 Let f, g be as in Theorem 1.1, 0 and , p, q be such that (1) 1 < q 2 < p < 2(N + 1)/N, > 0Then Equation (Pg) has infinitely many solutions. if , p, q satisfy that(3) 1< p < 2, 1< q < 2N/(N - 2), p g, 0, Then Equation (Pg) has at least one solution.Theorem 1.3 Let f, g be as in Theorem 1.1, 0 and , p, q be such that Then Equation (Pg) has at least one solution.Theorem 1.4 Let f, g be as in Theorem 1.1, 0 and , p, q be such thatThen Equation (Pg) has at least one solution.