On the fractional Lazer-McKenna conjecture with superlinear potential
The authors study superlinear problems involving the fractional Laplace operator \[ (-\Delta)^s u =|u|^{p} -\sigma\phi_1 \text{ in } \Omega, \text{ with } u >0 \text{ in } \Omega, \text{ and } u=0 \text{ in } \mathbb{R}^N\setminus\Omega \] where \(0<s<1\), \(1<p < \frac{N+2s}{N-2s}\), \(\Omega\) is a domain in \(\mathbb{R}^N\) with Lipschitz boundary, and \(\phi_1\) is the first positive eigenfunction of the fractional Laplacian with Dirichlet boundary conditions. By a Lyapunov-Schmidt reduction technique, as \(\sigma\) increases indefinitely, more and more multi-peaked solutions with peaks concentrated near local maximum points of \(\phi_1\) are obtained.
scientific article; zbMATH DE number 6991666
scientific article