Existence and Nonexistence Results for Classes of Singular Elliptic Problem

The singular semilinear elliptic problem in , in , on , is considered, where is a bounded domain with smooth boundary in , , and are three positive constants. Some existence or nonexistence results are obtained for solutions of this problem by the sub-supersolution method. 1. Introduction and Main Results In this paper, we study the existence or the nonexistence of solutions to the following singular semilinear elliptic problem where is a bounded domain with boundary for some , , and are three nonnegative constants. This problem arises in the study of non-Newtonian fluids, chemical heterogeneous catalysts, in the theory of heat conduction in electrically conducting materials (see [1–7] and their references). Many authors have considered this problem. For examples, when in , problem (1.1) was studied in [3, 8–11]; when in , problem (1.1) was considered in [12–14]. Particularly, when , it has been established in Zhang [14] that there exists such that problem (1.1) has at least one solution in for all and has no solution in if . After that Shi and Yao in [13] have also obtained the same results with and in . Recently, Ghergu and R？dulescu in [12] considered more general sublinear singular elliptic problem with . In this paper, we consider the case that , and may have zeros in . The following main results are obtained by the sub-supersolution method with restriction on the boundary in Cui [15]. Theorem 1.1. Suppose that , and . Assume that and , then there exists such that problem (1.1) has at least one solution and for all , and problem (1.1) has no solution in if . Moreover, problem (1.1) has a maximal solution which is increasing with respect to for all . Remark 1.2. Theorem 1.1 generalizes Theorem in [13] in coefficient of the singular term. Consequently, it also generalizes Theorem in [14]. Moreover, there are functions satisfying our Theorem 1.1 and not satisfying Theorem in [13]. For example, let where , and Certainly, this example does not satisfy Theorem in [12] yet. Theorem 1.3. Suppose that and in . If , problem (1.1) has no solution in for all and . Remark 1.4. Obviously, Theorem 1.3 is a generalization of Theorem in [14]. There are also functions satisfying our Theorem 1.3 and not satisfying Theorem in [14] and Theorem in [12]. For example, let where , is any positive constant and is the diameter of . 2. Proof of Theorems Consider the more general semilinear elliptic problem where the function is locally H？lder continuous in and continuously differentiable with respect to the variable . A function is called to be a subsolution of problem (2.1)