A concave-convex problem with a variable operator
Alexis Molino, Julio Daniel Rossi · arXiv (Cornell University) · 2017
We study the following elliptic problem $-A(u) = λu^q$ with Dirichlet boundary conditions, where $A(u) (x) = Δu (x) χ_{D_1} (x)+ Δ_p u(x) χ_{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the $p-$Laplacian (with $p>2$) in the rest of the domain, $D_2 $. We show that this problem exhibits a concave-convex nature for $1 λ^*$ and a minimal positive solution for $0