Nonlinear elliptic problems with a singular weight on the boundary

Dávila, Juan; Peral, Ireneo
July 2011
Calculus of Variations & Partial Differential Equations;Jul2011, Vol. 41 Issue 3/4, p567
Academic Journal
We study existence of solutions to with u = 0 on ∂Ω, where Ω is a smooth bounded domain in $${\mathbb {R}^N}$$ , N ≥ 3 with $${0\,\in\,\partial \Omega}$$ and $${1< p < \frac{N+2}{N-2}}$$ . The existence of solutions depends on the geometry of the domain. On one hand, if the domain is starshaped with respect to the origin there are no energy solutions. On the other hand, in dumbbell domains via a perturbation argument, the equation has solutions.


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