On a class of superlinear obstacle problems

Abstract

In this paper, we focus on the superlinear obstacle problem,$\{\begin{array}{cc}\Delta u=(1-u^{p-1}){\chi }_{\{u>0\}},& \mathrm{in}\Omega ,\\ u\ge 0,& \mathrm{in}\Omega ,\\ u\in W_0^{1,2}\left(\Omega \right),\end{array}$ where Ω is a smooth open bounded domain in $R^n,n\ge 2$, and the exponent $p>2$ satisfies the condition$2<p<\{\begin{array}{cc}{2}^{⁎},& \mathrm{as}n\ge 3,\\ +\infty ,& \mathrm{as}n=2,\end{array}$ with $2^{⁎}=\frac{2n}{n-2}$ is the Sobolev critical exponent of embedding $W_0^{1,2}\left(\Omega \right)\hookrightarrow L^q\left(\Omega \right)$ when $n\ge 3$. We prove the existence of a non-minimizing solution and establish the optimal regularity of solutions to the aforementioned equation. Furthermore, utilizing blowup analysis, we derive the regularity of the free boundary at regular points and characterize the structure of the singular set.