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Existence of solutions for elliptic problems involving the $(1,q)$-Laplacian operator and a discontinuous superlinear nonlinearity
[Submitted on 18 Jun 2026] · 2026-06-19 · via math updates on arXiv.org

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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 $p\to1^+$. As a consequence, we prove the existence of a nontrivial and nonnegative solution belonging to $W^{1,p}_0(\Omega)$, in an appropriate weak sense. Moreover, we investigate the asymptotic behavior of the solutions as $\beta\to0^+$, showing that the family of solutions converges to a solution of the limit problem without discontinuity.

Submission history

From: Marcos Antonio Viana Costa [view email]
[v1] Thu, 18 Jun 2026 13:32:26 UTC (18 KB)

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