Solutions in Lebesgue spaces to nonlinear elliptic equations with subnatural growth terms

A. Seesanea ; I. E. Verbitsky

Algebra i analiz, Tome 31 (2019) no. 3, pp. 216-238.

Résumé

The paper is devoted to the existence problem for positive solutions $ {u \in L^{r}(\mathbb{R}^{n})}$, $ 0$, to the quasilinear elliptic equation $$ -\Delta _{p} u = \sigma u^{q} \text { in } \mathbb{R}^n $$ in the subnatural growth case $ 0$, where $ \Delta _{p}u = \mathrm {div}( \vert\nabla u\vert^{p-2} \nabla u )$ is the $ p$-Laplacian with $ 1$, and $ \sigma $ is a nonnegative measurable function (or measure) on $ \mathbb{R}^n$. The techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators in place of $ \Delta _{p}$ such as the $ \mathcal {A}$-Laplacian $ \mathrm {div} \mathcal {A}(x,\nabla u)$, or the fractional Laplacian $ (-\Delta )^{\alpha }$ on $ \mathbb{R}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $ \mathrm {div} (\mathcal {A} \nabla u)$ on an arbitrary domain $ \Omega \subseteq \mathbb{R}^n$ with a positive Green function.

Détail

Keywords: quasilinear elliptic equation, measure data, $p$-Laplacian, fractional Laplacian, Wolff potential, Green function.

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