主讲人：李工宝 教授（华中师范大学）

时  间：5月16日（周四）14：10-15：10

地  点：威尼斯欢乐娱人城1099北二区教学楼 510 教室

备  注：In this talk, I will present some results in my recent joint work with Yi He on the existence and concentration of weak solutions to the $p$-Laplacian type elliptic problem \[ \left\{ \begin{gathered} - {\varepsilon ^p}{\Delta _p}u + V\left( z \right){\left| u \right|^{p - 2}}u - f\left( u \right) = 0{\text{ in }}\Omega , \hfill \\ u = 0{\text{ on }}\partial \Omega ,u > 0{\text{ in }}\Omega ,N > p > 2 \hfill \\ \end{gathered} \right. \] where $\Omega $ is a domain in ${R^N}$, possibly unbounded, with empty or smooth boundary, $\varepsilon $ is a small positive parameter, $f \in {C^1}\left( {{R^ + },R} \right)$ is of subcritical, $V:{R^N} \to R$ is a locally H\"{o}lder continuous function which is bounded from below, away from zero, such that $\mathop {\inf }\limits_\Lambda V < \mathop {\min }\limits_{\partial \Lambda } V$ for some open bounded subset $\Lambda $ of $\Omega $. We prove that there is an ${\varepsilon _0}> 0$ such that for any $\varepsilon \in \left( {0,{\varepsilon _0}} \right]$, the above mentioned problem possesses a weak solution ${u_\varepsilon }$ with exponential decay. Moreover, ${u_\varepsilon }$ concentrates around a minimum point of the potential $V$ in $\Lambda $. Our result generalizes a similar result in [M. del Pino, P. L. Felmer: Cal. Var. 4(1996), 121-137.] for semilinear elliptic equations to the $p$-Laplacian type problem.