Existence and some properties of solutions for degenerate elliptic equations with exponent variable

Abstract

In this paper, we study degenerate elliptic equations with variable exponents when a perturbation term satisfies the Ambrosetti–Rabinowitz condition and does not satisfy the Ambrosetti–Rabinowitz condition. For the first case, we employ the standard Mountain Pass theorem to give the existence of solutions. For the second case, we use Browder’s theorem for monotone operators to show the unique existence of a solution when the perturbation term is decreasing with respect to a function variable. A priori bound and nonnegativeness of solutions are also given. We emphasize that the log-Hölder continuous condition is not required.