On existence of solutions for a class of problems involving a nonlinear operator (Q2782920)

The main result is concerned with the nonlocal sublinear Dirichlet problem \(-M(\int_\Omega |\nabla u|^2 dx)\) \(\Delta u=u^q\) in \(\Omega\), \(u=0\) on \(\partial\Omega\). Here \(0<q<1\), and the function \(M:\mathbb{R}\to \mathbb{R}\) is subject to the following conditions: \(M\) is bounded from below by a positive constant; \(M\) is non-increasing; \(\mathbb{R}\ni t\mapsto M(t^2)t\in\mathbb{R}\) is increasing and surjective; \(t\mapsto(M(t^2))^{1/(q-1)}t\) is injective on \([0, \infty)\). Under these conditions the above Dirichlet problem has a unique positive solution. This result is proved with help of a comparison principle and a sub-/supersolution-method. The assumptions are satisfied e.g. for \(M(t)=\exp (-t)+C\), where \(C>0\) is a sufficiently large constant.