POSITIVE SOLUTIONS TOp(x)-LAPLACIAN–DIRICHLET PROBLEMS WITH SIGN-CHANGING NON-LINEARITIES

Abstract

AbstractConsider thep(x)-Laplacian–Dirichlet problem with sign-changing non-linearity of the formwhere Ω ⊂ ℝNis a bounded domain,p∈C0(Ω) and infx∈Ωp(x) > 1,m∈L∞(Ω) is non-negative,f: ℝ → ℝ is continuous andf(0) > 0, the coefficienta∈L∞(Ω) is sign-changing in (Ω). We give some sufficient conditions to assure the existence of a positive solution to the problem for sufficiently small λ > 0. Our results extend the corresponding results established in thep-Laplacian case to thep(x)-Laplacian case.