Positive constrained minimizers for supercritical problems in the ball

Source abstract

We provide a sufficient condition for the existence of a positive solution to $$− Δ u + V ( | x | ) u = u p in B 1 , -\Delta u+V(|x|) u=u^p \quad \hbox { in } B_1,$$ when p p is large enough. Here B 1 B_1 is the unit ball of R n \mathbb {R}^n , n ≥ 2 n\ge 2 , and we deal with both Neumann and Dirichlet homogeneous boundary conditions. The solution turns out to be a constrained minimum of the associated energy functional. As an application we show that in case V ( | x | ) ≥ 0 V(|x|)\geq 0 , V ≢ 0 V\not \equiv 0 is smooth and p p is sufficiently large, and the Neumann problem always admits a solution.