Abstract
This paper investigates the following p(x)‐Laplacian equations with exponential nonlinearities: −Δp(x)u + ρ(x)ef(x,u) = 0 in Ω, u(x)→+∞ as d(x, ∂Ω) → 0, where −Δp(x)u = −div(|∇u|p(x)−2∇u) is called p(x)‐Laplacian, ρ(x) ∈ C(Ω). The asymptotic behavior of boundary blow‐up solutions is discussed, and the existence of boundary blow‐up solutions is given.
Highlights
The study of differential equations and variational problems with nonstandard p x -growth conditions is a new and interesting topic
Because of the nonhomogeneity of p x -Laplacian, p x -Laplacian problems are more complicated than those of p-Laplacian ones see 10 ; another difficulty of this paper is that f x, u cannot be represented as h x f u
In order to deal with p x -Laplacian problems, we need some theories on the spaces Lp x Ω, W1,p x Ω and properties of p x -Laplacian, which we will use later see 6, 11
Summary
The study of differential equations and variational problems with nonstandard p x -growth conditions is a new and interesting topic. On the existence of solutions for p x -Laplacian equation Dirichlet problems in bounded domain, we refer to 7, 9, 15, 18. Our aim is to give the asymptotic behavior and the existence of boundary blow-up solutions for problem P. U can be represented as h x f u , on the boundary blow-up solutions for the following p-Laplacian equations p is a constant :. On the boundary blow-up solutions for the following p-Laplacian equations with exponential nonlinearities p is a constant :. −Δpu eh x f u 0, in Ω, 1.7 we refer to 20–22 , but the results on the boundary blow-up solutions for p x -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 p x -Laplacian equations:. Because of the nonhomogeneity of p x -Laplacian, p x -Laplacian problems are more complicated than those of p-Laplacian ones see 10 ; another difficulty of this paper is that f x, u cannot be represented as h x f u
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