Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources

Abstract

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equationDue to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent α(x), some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of α(x).

Nonlinear elliptic equations with variable exponents involving singular nonlinearity

“…In case without the lower-order term in (1) (i.e., b(x) = 0) and the exponent p(x) ≡ p, the problem (1), have been treated in [11], under the hypothesis f ∈ L m (Ω) (m 1). If m = 1 and 0 < γ − γ(x) γ + < 1 the authors proved that the solution belongs to W 1,q 0 (Ω), where q = N (p+γ − −1) N +γ − −1 .…”

Nonlinear Elliptic Equations With Variable Exponents Involving Singular Nonlinearity

“…When the lower-order term does not appear in (1.1) (i.e., b(x) = 0) and the exponent p(x) ≡ p, the problem (1.1), have been treated in [13], under the hypothesis f ∈ L m (Ω) (m ≥ 1). If m = 1 and 0 < γ − ≤ γ(x) ≤ γ + < 1 the authors proved that the solution belongs to W 1,q 0 (Ω), where q = N (p+γ − −1) N +γ − −1 .…”

Nonlinear singular problems with indefinite potential term

“…It is also related to the work of Papageorgiou & Smyrlis [17] and Papageorgiou & Winkert [19], where the differential operator is the p-Laplacian, ξ ≡ 0 and no concave terms are allowed. Singular p-Laplacian equations with no potential term and reactions of special form were considered by Chu, Gao & Sun [2], Giacomoni, Schindler & Takač [5], Li & Gao [10], Mohammed [12], Perera & Zhang [20], and Papageorgiou, Rȃdulescu & Repovš [14].…”