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 and MONTENEGRO, Marcos. The effect of diffusion on critical quasilinear elliptic problems. Topological Methods in Nonlinear Analysis. Online. 12 April 2016. Vol. 43, no. 2, pp. 517 - 534. [Accessed 28 March 2024].
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