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The Dirichlet problem for singular elliptic equations with general nonlinearities

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Abstract

In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle - \Delta _{1} u = h(u)f &{} \text {in}\, \Omega , \\ u\ge 0&{} \text {in}\ \Omega ,\\ u=0 &{} \text {on}\ \partial \Omega \,. \end{array}\right. } \end{aligned}$$

Here \(\Delta _{1} \) is the 1-Laplace operator, \(\Omega \) is a bounded open subset of \(\mathbb {R}^N\) with Lipschitz boundary, h(s) is a continuous function which may become singular at \(s=0^{+}\), and f is a nonnegative datum in \(L^{N,\infty }(\Omega )\) 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.

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Authors and Affiliations

  1. Dipartimento di Scienze di Base e Applicate per l’ Ingegneria, “Sapienza” Università di Roma, Via Scarpa 16, 00161, Rome, Italy

    Virginia De Cicco, Daniela Giachetti & Francesco Petitta

  2. Istituto Nazionale di Alta Matematica (INdAM), Dipartimento di Scienze di Base e Applicate per l’ Ingegneria, “Sapienza” Università di Roma, Via Scarpa 16, 00161, Rome, Italy

    Francescantonio Oliva

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  1. Virginia De Cicco
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  2. Daniela Giachetti
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  3. Francescantonio Oliva
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  4. Francesco Petitta
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Correspondence to Francesco Petitta.

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Communicated by J. Ball.

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De Cicco, V., Giachetti, D., Oliva, F. et al. The Dirichlet problem for singular elliptic equations with general nonlinearities. Calc. Var. 58, 129 (2019). https://doi.org/10.1007/s00526-019-1582-4

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