Existence and Asymptotic Behavior of Boundary Blow-Up Solutions for Weighted p(x)-Laplacian Equations with Exponential Nonlinearities
نویسندگان
چکیده
and Applied Analysis 3 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: −Δp x u f x, u 0, in Ω, u x −→ ∞, as d x, ∂Ω −→ 0, 1.8 on the condition that f x, · satisfies polynomial growth condition. If p x is a function, the typical form of P is the following: −Δp x u ρ x e|u| q x −2u 0, 1.9 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 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 . 2. Preliminary In order to deal with p x -Laplacian problems, we need some theories on the spaces L x Ω , W1,p x Ω and properties of p x -Laplacian, which we will use later see 6, 11 . Let L x Ω { u | u is a measurable real-valued function, ∫Ω |u x | x dx < ∞ } . 2.1 We can introduce the norm on L x Ω by |u|p x inf { λ > 0 | ∫ Ω ∣ ∣ ∣ ∣ u x λ ∣ ∣ ∣ ∣ p x dx ≤ 1 } . 2.2 The space L x Ω , | · |p x becomes a Banach space. We call it generalized Lebesgue space. The space L x Ω , | · |p x is a separable, reflexive, and uniform convex Banach space see 6, Theorems 1.10, 1.14 . The space W1,p x Ω is defined by W1,p x Ω { u ∈ L x Ω | |∇u| ∈ L x Ω } , 2.3 4 Abstract and Applied Analysis and it can be equipped with the norm ‖u‖ |u|p x |∇u|p x , ∀u ∈ W1,p x Ω . 2.4 W 1,p x 0 Ω is the closure of C ∞ 0 Ω in W 1,p x Ω . W1,p x Ω and W x 0 Ω are separable, reflexive, and uniform convex Banach spaces see 6, Theorem 2.1 . If u ∈ W x loc Ω ∩ C Ω , u is called a blow-up solution of P when it satisfies
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