Problem with Critical Sobolev Exponent and with Weight

Abstract

The authors consider the problem: −div(p∇u) = u ^q−1 +λu, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝ ⁿ, n ≥ 3, $p:\ifmmode\expandafter\bar\else\expandafter\=\fi{\Omega } \to \mathbb{R}$ is a given positive weight such that $p \in H^{1} {\left( \Omega \right)} \cap C{\left( {\ifmmode\expandafter\bar\else\expandafter\=\fi{\Omega }} \right)},\lambda $ is a real constant and $q = \frac{{2n}}{{n - 2}}$, and study the effect of the behavior of p near its minima and the impact of the geometry of domain on the existence of solutions for the above problem.

Keywords

Critical Sobolev exponent / Variational methods / 35J20 / 35J25 / 35J60

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