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MiS Preprint
51/2002
On a nonlinear elliptic equation arising in a free boundary problem
Guofang Wang and Dong Ye
Abstract
Let $p^*= n/(n-2)$ and $n \geq 3$. In this paper, we first classify all non-constant solutions of \[ \left\{\begin{array}{rl} -\Delta u &\!\! = u_+^{p^*} \quad \hbox{ in } {\Bbb R}^n,\\ \int_{{\Bbb R}^n} u_+^{p^*} dx &\!\! < \infty. \end{array}\right.\] We then establish a $\sup + \inf$ and a Moser-Trudinger type inequalities for the equation $-\Delta u = u_+^{p^*}$. Our results illustrate that this equation is much closer to the Liouville problem $-\Delta u = e^u$ in dimension two than the usual critical exponent equation, namely $-\Delta u = u^{\frac{n+2}{n-2}}$ is.
