Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

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 [1–9] and their references). But most of such studies have been concerned with equations of the type involving sublinear nonlinearity (see [3–6, 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