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In this work, we study a quasilinear elliptic problem involving the 1-Laplacian operator, with a discontinuous, superlinear and subcritical nonlinearity involving the Heaviside function $H(\cdot - \beta )$. Our approach is based on an analysis of the associated p-Laplacian problem, followed by a thorough analysis of the asymptotic behaviour or such solutions as $p \to 1^+$. We study also the asymptotic behaviour of the solutions, as $\beta \to 0^+$ and we prove that it converges to a solution of the original problem, without the discontinuity in the nonlinearity.

For a class of elliptic equations in the entire space and with nonlinear terms having a possibly uncountable (but of Lebesgue measure zero) set of discontinuities, the existence of strong solutions is established. Two simple applications are then developed. The approach taken is strictly based on set-valued analysis and fixed-points arguments.

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On a quasilinear elliptic problem involving the 1-Laplacian operator and a discontinuous nonlinearity

Elliptic problems in ℝN with discontinuous nonlinearities

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On a quasilinear elliptic problem involving the 1-Laplacian operator and a discontinuous nonlinearity

Elliptic problems in ℝN with discontinuous nonlinearities