Existence of Positive Solutions for p(x)-Laplacian Equations with a Singular Nonlinear Term

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 ⊂ RN . 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 interest.

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