Existence and Asymptotic Behavior of Boundary Blow-Up Solutions for Weighted -Laplacian Equations with Exponential Nonlinearities

This paper investigates the following -Laplacian equations with exponential nonlinearities: in , as , where is called -Laplacian, . The asymptotic behavior of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given. 1. Introduction The study of differential equations and variational problems with nonstandard -growth conditions is a new and interesting topic. On the background of this class of problems, we refer to [1–3]. Many results have been obtained on this kind of problems, for example, [4–18]. On the regularity of weak solutions for differential equations with nonstandard -growth conditions, we refer to [4, 5, 8]. On the existence of solutions for -Laplacian equation Dirichlet problems in bounded domain, we refer to [7, 9, 15, 18]. In this paper, we consider the following -Laplacian equations with exponential nonlinearities where and is a bounded radial domain ( ). Our aim is to give the asymptotic behavior and the existence of boundary blow-up solutions for problem (P). Throughout the paper, we assume that , , and satisfy the following. (H1) is radial and satisfies (H2) is radial with respect to , is increasing, and for any . (H3)？ is continuous and satisfies where , are positive constants and . (H4) is a radial nonnegative function, and there exists a constant such that where and are positive constants and and are Lipschitz continuous on , which satisfy for any . The operator is called -Laplacian. Specifically, if (a constant), (P) is the well-known -Laplacian problem. If can be represented as , on the boundary blow-up solutions for the following -Laplacian equations ( is a constant): we refer to [19–26], and the following generalized Keller-Osserman condition is crucial but the typical form of -Laplacian equation is and there are some differences between the results of (1.4) and (1.6) (see [16]). On the boundary blow-up solutions for the following -Laplacian equations with exponential nonlinearities ( is a constant): we refer to [20–22], but the results on the boundary blow-up solutions for -Laplacian equations are rare (see [16]). In [16], the present author discussed the existence and asymptotic behavior of boundary blow-up solutions for the following -Laplacian equations: on the condition that satisfies polynomial growth condition. If is a function, the typical form of (P) is the following: and the method to construct subsolution and supersolution in [16] cannot give the exact asymptotic behavior of solutions for (P). Our results partially generalized the results of [20–22]. Because of the