In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form {-Δ1u=h(u)finΩ,u≥0inΩ,u=0on∂Ω.Here Δ 1 is the 1-Laplace operator, Ω is a bounded open subset of RN with Lipschitz boundary, h(s) is a continuous function which may become singular at s= 0 +, and f is a nonnegative datum in LN,∞(Ω) with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and f> 0. As a preparatory tool for our method a general theory for the same problem involving the p-Laplacian (with p> 1) as principal part is also established. The main assumptions are further discussed in order to show their optimality.

The Dirichlet problem for singular elliptic equations with general nonlinearities / De Cicco, V.; Giachetti, D.; Oliva, F.; Petitta, F.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 58:4(2019). [10.1007/s00526-019-1582-4]

The Dirichlet problem for singular elliptic equations with general nonlinearities

Oliva F.;
2019

Abstract

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form {-Δ1u=h(u)finΩ,u≥0inΩ,u=0on∂Ω.Here Δ 1 is the 1-Laplace operator, Ω is a bounded open subset of RN with Lipschitz boundary, h(s) is a continuous function which may become singular at s= 0 +, and f is a nonnegative datum in LN,∞(Ω) with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and f> 0. As a preparatory tool for our method a general theory for the same problem involving the p-Laplacian (with p> 1) as principal part is also established. The main assumptions are further discussed in order to show their optimality.
2019
The Dirichlet problem for singular elliptic equations with general nonlinearities / De Cicco, V.; Giachetti, D.; Oliva, F.; Petitta, F.. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - 58:4(2019). [10.1007/s00526-019-1582-4]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11588/817517
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