Positive solutions in semilinear critical problems for polyharmonic operators
The author deals with the semilinear polyharmonic problem \[ \begin{gathered} (-\Delta)^k u= |u|^{s-2} u+ f(x,u)\quad\text{in }\Omega,\\ u> 0\quad\text{in }\Omega,\\ u= (-\Delta)u=\cdots=(-\Delta)^{k-1}= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(k\in\mathbb{N}\) and \(\Omega\subset\mathbb{R}^N\) \((N\geq 2k+1)\) is a regular bounded domain in \(\mathbb{R}^N\); \(s= {2N\over N-2k}\). The author studies the existence of solutions to (1) for general bounded domains under appropriate assumptions minimizing on some infinite-dimensional Finsler manifold. Moreover, he studies a concentration phenomenon.
scientific article; zbMATH DE number 2164963
scientific article