Education Commons^™

Existence Of Positive Solutions For P(X)-Laplacian Equations With A Singular Nonlinear Term, Jingjing Liu, Qihu Zhang, Chunshan Zhao

Department of Mathematical Sciences Faculty Publications

In this article, we study the existence of positive solutions for the p(x)-Laplacian Dirichlet problem −∆_p(x)u = λf(x, u) in a bounded domain Ω ⊂ R^N. The singular nonlinearity term f is allowed to be either f(x, s) → +∞, or f(x, s) → +∞ as s → 0+ for each x ∈ Ω. Our main results generalize the results in [15] from constant exponents to variable exponents. In particular, we give the asymptotic behavior of solutions of a simpler equation which is useful for finding supersolutions of differential equations with variable exponents, which is of independent …

On The Boundary Blow-Up Solutions Of P(X)-Laplacian Equations With Gradient Terms, Yuan Liang, Qihu Zhang, Chunshan Zhao

Department of Mathematical Sciences Faculty Publications

In this paper we investigate boundary blow-up solutions of the problem

⎧⎩⎨⎪⎪−△p(x)u+f(x,u)=ρ(x,u)+K(|x|)|∇u|δ(|x|) in Ω, u(x)→+∞ as d(x, ∂Ω)→0,

where −△p(x)u=−div(|∇u|p(x)−2∇u) is called p(x)-Laplacian. The existence of boundary blow-up solutions is proved and the singularity of boundary blow-up solution is also given for several cases including the case of ρ(x,u) being a large perturbation (namely, ρ(x,u(x))f(x,u(x))→1 as x→∂Ω). In particular, we do not have the comparison principle.

Multiple Solutions Of A P(X)-Laplacian Equation Involving Critical Nonlinearities, Yuan Liang, Xianbin Wu, Qihu Zhang, Chunshan Zhao

Department of Mathematical Sciences Faculty Publications

In this paper, we consider the existence of multiple solutions for the following p(x)-Laplacian equations with critical Sobolev growth conditions

−div(|∇u| p(x)−2 ∇u) + |u| ^p(x)−2 u = f(x, u) in Ω,

u = 0 on ∂Ω.

We show the existence of infinitely many pairs of solutions by applying the Fountain Theorem and the Dual Fountain Theorem respectively. We also present a variant of the concentration-compactness principle, which is of independent interest.