Existence, non-existence and degeneracy of limit solutions to $p$-Laplace problems involving Hardy potentials as $p\to 1^{+}$. The case of a critical drift

Abstract

In this paper we analyze the asymptotic behaviour as $p\to 1^{+}$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -Δ_pu&=&λ|\nabla u|^{p-2}\nabla u\cdot\frac{x}{|x|^2}+ f&\quad \mbox{ in } Ω,\\ u_p&=&0 &\quad \mbox{ on }\partialΩ, \end{array}\right. $$ where $\Omega$ is a bounded open subset of ${\mathbb{R}}^N$ with Lipschitz boundary containing the origin, $\lambda \in \mathbb{R}$, and $f$ is a nonnegative datum in $L^{N,\infty }(\Omega )$. As a consequence, under suitable smallness assumptions on $f$ and $\lambda$, we show sharp existence results of bounded solutions to the Dirichlet problems $$\{\begin{array}{llc}-{\Delta }_{1}u=\lambda \frac{Du}{|Du|}\cdot \frac{x}{|x{|}^2}+f& \text{in}\Omega ,u=0& \text{on}\partial \Omega ,\end{array}$$ where ${\Delta }_{1}u=div\left(\frac{Du}{|Du|}\right)$ is the $1$-Laplacian operator. The case of a generic drift term in $L^{N,\infty }(\Omega )$ is also considered. Explicits examples are given in order to show the optimality of the main assumptions on the data.

arXiv admin note: text overlap with arXiv:2401.15406