The effect of diffusion on critical quasilinear elliptic problems
Keywords
Non-uniformly elliptic operators, critical Sobolev exponent, best constant, Hardy-Sobolev inequalityAbstract
We discuss the role of the diffusion coefficient $a(x)$ on the existence of a positive solution for the quasilinear elliptic problem involving critical exponent $$ \cases - \text{div}( a(x) |\nabla u|^{p-2} \nabla u) = u^{p^* - 1} + \lambda u^{p-1} & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega,\ \endcases $$ where $\Omega$ is a smooth bounded domain in $\R^n$, $n \geq 2$, $1 < p < n$, $p^* = np/(n-p)$ is the critical exponent from the viewpoint of Sobolev embedding, $\lambda$ is a real parameter and $a\colon \overline{\Omega} \rightarrow \R$ is a positive continuous function. We prove that if the function $a(x)$ has an interior global minimum point $x_0$ of order $\sigma$, then the range of values $\lambda$ for which the problem above has a positive solution relies strongly on $\sigma$. We discover in particular that the picture changes drastically from $\sigma > p$ to $\sigma \leq p$. Some sharp answers are also provided.Downloads
Published
2016-04-12
How to Cite
1.
DE MOURA, Renato José & MONTENEGRO, Marcos. The effect of diffusion on critical quasilinear elliptic problems. Topological Methods in Nonlinear Analysis [online]. 12 April 2016, T. 43, nr 2, s. 517–534. [accessed 26.1.2023].
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