| Abstract: |
| We present the uniqueness of nontrivial solutions to the problem
\begin{equation*}
\left\{ \begin{array}{ll}
- \Delta_p u = \sigma u^q + \mu, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \
\displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0,
\end{array}
\right.
\end{equation*} in the sub-natural growth case $0\lt q\lt p-1$,
where $\mu, \sigma$ are nonnegative locally finite measures in $\mathbb{R}^n$ absolutely continuous with respect
to the $p$-capacity. Here $\Delta_p u:={\rm div}(|\nabla u|^{p-2}\nabla u)$, $1\lt p\lt \infty$, is the $p$-Laplace operator.
The uniqueness is obtained in the class of \emph{reachable} solutions, and moreover, if the condition $\displaystyle{\liminf_{|x|\rightarrow \infty}}\, u = 0$
is replaced by the stronger condition $\displaystyle{\lim_{|x|\rightarrow \infty}}\, u = 0$, then such uniqueness is obtained in the larger class of $p$-superharmonic solutions.
This talk is based on joint work with Igor E. Verbitsky. |
|