Existence and Uniqueness of Solutions to Singular p-Laplace Equations of Kirchhoff Type

Li, Qingwei

Canadian mathematical bulletin, Tome 63 (2020) no. 3, pp. 677-691

Résumé

In this paper, we study both the existence and uniqueness of nonnegative solutions for the nonlocal $p$-Laplace equation with singular term $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}-B\Bigl(\frac{1}{p}\int _{\unicode[STIX]{x1D6FA}}|\unicode[STIX]{x1D6FB}u|^{p}\text{d}x\Bigr)\unicode[STIX]{x1D6E5}_{p}u=\frac{h(x)}{u^{\unicode[STIX]{x1D6FE}}}+k(x)u^{q},\quad & x\in \unicode[STIX]{x1D6FA},\\ u>0,\quad & x\in \unicode[STIX]{x1D6FA},\\ u=0,\quad & x\in \unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$ where $\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}(N\geqslant 1)$ is a bounded domain with smooth boundary $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$, $h\in L^{1}(\unicode[STIX]{x1D6FA})$, $h>0$ almost everywhere in $\unicode[STIX]{x1D6FA}$, $k\in L^{\infty }(\unicode[STIX]{x1D6FA})$ is a non-negative function, $B:[0,+\infty )\rightarrow [m,+\infty )$ is continuous for some positive constant $m$, $p>1$, $0\leqslant q\leqslant p-1$, and $\unicode[STIX]{x1D6FE}>1$. A “compatibility condition” on the couple $(h(x),\unicode[STIX]{x1D6FE})$ will be given for the problem to admit at least one solution. To be a little more precise, it is shown that the problem admits at least one solution if and only if there exists a $u_{0}\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$ such that $\int _{\unicode[STIX]{x1D6FA}}h(x)u_{0}^{1-\unicode[STIX]{x1D6FE}}\text{d}x<\infty$. When $k(x)\equiv 0$, the weak solution is unique.

Détail

DOI : 10.4153/S0008439519000766

Mots-clés : nonlocal, singular, existence, uniqueness, compatibility condition

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