%0 Journal Article
%T Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities
%A Chunmei Yuan
%A Shujuan Guo
%A Kaiyu Tong
%J Mathematical Problems in Engineering
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/640841
%X This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities in , in , on , where is a real parameter, . is a bounded domain in , and and are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions. 1. Introduction In this paper, we consider the elliptic problems with sublinear and superlinear nonlinearities where £¿£¿is a real parameter,£¿£¿ . £¿ is a bounded domain in ,£¿and£¿ and are some given functions which satisfies the following assumptions:( ) ,£¿£¿ ,£¿£¿ ,£¿£¿where ,£¿£¿ are positive constants, or( ) ,£¿ ,£¿£¿ ,£¿£¿where is a positive constant. For convenience, we denote with hypothesis or by and , respectively. Such problems occur in various branches of mathematical physics and population dynamics, and sublinear analogues or superlinear analogues of have been considered by many authors in recent years (see [1¨C9] and their references). But most of such studies have been concerned with equations of the type involving sublinear nonlinearity (see [3¨C6, 8, 9]), with only few references dealing with the elliptic problems with sublinear and superlinear nonlinearities. In [1], Ambrosetti et al. deal with the analogue of with . It is known from [2] that there exist , such that problem has a solution if £¿£¿and has no solution if ,£¿£¿provided on . Our goal in this paper is to show how variational method and super-subsolution method can be used to establish some existence results of problem . We work on the Sobolev space £¿equipped with the norm£¿£¿ . £¿£¿For £¿we define £¿£¿by Let be the first eigenvalue of £¿£¿denotes the corresponding eigenfunction satisfying£¿£¿ . £¿ ,£¿£¿ , denotes Lebesgue spaces, and the norm in is denoted by . 2. The Existence of Positive Solution of It is well known that Define£¿£¿ ; £¿£¿from (2.1) we know£¿£¿ , £¿so we can split the domain into two parts: and , where . Let£¿£¿ ; £¿we obtain that£¿£¿ £¿£¿by the positivity of £¿£¿in£¿£¿ , £¿and£¿£¿ £¿£¿is nonempty when£¿£¿ £¿£¿is small enough. Theorem 2.1. Let£¿£¿ ,£¿£¿ satisfy assumption , £¿and£¿£¿ , where£¿£¿ is the limiting exponent in the Sobolev embedding. Then there exists a constant £¿£¿such that possesses at least a weak positive solution £¿for£¿£¿ . Proof. Let£¿£¿ £¿denote the positive solution of the following equation: Here and hereafter we use the following notations: , £¿£¿ , £¿£¿ . Since , for all , £¿there exists£¿£¿ £¿£¿satisfying Observing that£¿£¿ , £¿as a consequence, the function£¿£¿ £¿£¿verifies and hence it is a supersolution of . £¿Let£¿£¿ ,£¿£¿ ,£¿£¿ . For , we have or . We will discuss it from two
%U http://www.hindawi.com/journals/mpe/2010/640841/