Blowing Up Solutions to Slightly Sub- or Super-Critical Lane-Emden Systems with Neumann Boundary Conditions (Analysis in Theory and Applications, 2025)
We prove that, for some suitable smooth bounded domain, there exists asolution to the following Neumann problem for the Lane-Emden system:where $Ω$ is some smooth bounded domain in $\mathbb{R}^N,$ $N ≥ 4,$ $μ > 0,$ $α > 0,$ $β > 0$ areconstants and $ε e 0$ is a small number. We show that there exists a solution to theslightly supercritical problem for $ε > 0,$ and for $ε < 0,$ there also exists a solution to theslightly subcritical problem if the domain is not convex.Comparing with the single elliptic equations, the challenges and novelty are manifested in the construction of good approximate solutions characterizing the boundarybehavior under Neumann boundary conditions, in which process, the selection of therange of nonlinear coupling exponents and the weighted Sobolev spaces requires elaborate discussion.
Keywords: Lane-Emden system; Neumann problem; blow up solutions; reduction method.