The Dirichlet problem for singular elliptic equations with general nonlinearities


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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Correspondence to Francesco Petitta.

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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).

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Mathematics Subject Classification

  • 35J60
  • 35J75
  • 34B16
  • 35R99
  • 35A02