The generalized logistic equation with indefinite weight driven by the square root of the Laplacian (Q2924851)

The paper deals with the Dirichlet problem NEWLINE\[NEWLINE \begin{cases} \sqrt{-\Delta}u(x)=\lambda \big( \beta(x) u(x)-g(x,u(x)\big) & \mathrm{in}\;\Omega,\\ u(x)=0 & \mathrm{on}\;\partial\Omega \end{cases} NEWLINE\]NEWLINE over a bounded domain \(\Omega\subset \mathbb{R}^N,\) \(N\geq2,\) where \(\beta(x)\) is a sign-changing measurable weight. The authors prove a bifurcation result for the problem considered via regularity estimates when the weight belongs only to certain Lebesgue spaces.