Existence of solutions for elliptic problems involving the $(1,q)$-Laplacian operator and a discontinuous superlinear nonlinearity

Abstract

In this paper, we study a class of quasilinear elliptic problems involving the $(1,q)-$Laplacian operator and a discontinuous superlinear nonlinearity governed by the Heaviside function. The main difficulty of the problem arises from the presence of the $1$-Laplacian operator, whose natural setting is the Space of Functions of Bounded Variation. Our approach is based on an approximation method involving $(p,q)-$Laplacian problems as $p\to 1^{+}$. As a consequence, we prove the existence of a nontrivial and nonnegative solution belonging to $W_0^{1,p}(\Omega )$, in an appropriate weak sense. Moreover, we investigate the asymptotic behavior of the solutions as $\beta \to 0^{+}$, showing that the family of solutions converges to a solution of the limit problem without discontinuity.