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

Abstract

In this paper we analyze the asymptotic behaviour as $p\to 1^{+}$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -Δ_p u_p&=&\fracλ{|x|^p}|u_p|^{p-2}u_p+f&\quad \mbox{ in } Ω,\\ u_p&=&0 &\quad \mbox{ on }\partialΩ, \end{array}\right. $$ where $\Omega$ is a bounded open subset of ${\mathbb{R}}^N$ with Lipschitz boundary, $\lambda \in {\mathbb{R}}^{+}$, and $f$ is a nonnegative datum in $L^{N,\infty }(\Omega )$. Under sharp smallness assumptions on the data $\lambda$ and $f$ we prove that $u_p$ converges to a suitable solution to the homogeneous Dirichlet problem $$\{\begin{array}{rclr}-{\Delta }_{1}u& =& \frac{\lambda }{|x|}\mathrm{Sgn}(u)+f& \text{in}\Omega ,\\ u& =& 0& \text{on}\partial \Omega ,\end{array}$$ where ${\Delta }_{1}u=\mathrm{div}\left(\frac{Du}{|Du|}\right)$ is the $1$-Laplace operator. The main assumptions are further discussed through explicit examples in order to show their optimality.