Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions

Abstract In the present paper, we study the existence as well as the non-existence of some positive solutions for the equation −Δp(x) u = λ k(x) uq ± h(x) ur under Robin boundary condition in a regular open bounded domain Ω of ℝN, N ≥ 2. Δp(x) is the p(x)-Laplacian operator where p ∈ C1(Ω) and p > 1. Our proofs are based on the sub solution-super solution method and also on variational arguments.