Global solution branches for \(p\)-Laplacian boundary value problems
The authors study global continua of positive solutions to the Dirichlet problem \[ \begin{cases} -\Delta_p u= \lambda (1+u^q) & \text{in}\;\Omega\subset \mathbb R^n,\\ u=0 & \text{on}\;\partial\Omega. \end{cases} \] It is shown that, for subcritical values of \(q,\) there exists an unbounded continuum of positive solutions such that for every \(\lambda\in(0,\lambda^*)\) there are at least two solutions but for \(\lambda>\lambda^*\) no solution exists. Contrary, uniqueness holds for small positive \(\lambda\) when \(q\) is supercritical. The multiplicity result is proved by means of topological degree arguments combined with recent a~priori bounds, while the uniqueness for supercritical \(q\) follows from the Pokhozhaev identity and new weighted Poincaré-type inequalities.
scientific article; zbMATH DE number 2183183
scientific article