Abstract.
We investigate the existence of ground state solutions to the Dirichlet problem \(-\div(|x|^\alpha\nabla u)=|u|^{2^*_\alpha-2}u\) in \(\Omega\), u = 0 on \(\partial\Omega\), where \(\alpha\in(0,2)\), \(2^*_\alpha={2n\over n-2+\alpha}\) and \(\Omega\) is a domain in \({\bf R}^n\). In particular we prove that a non negative ground state solution exists when the domain \(\Omega\) is a cone, including the case \(\Omega={\bf R}^n\). Moroever, we study the case of arbitrary domains, showing how the geometry of the domain near the origin and at infinity affects the existence or non existence of ground state solutions.
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Received April 16, 1998 / Accepted June 24, 1998
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Caldiroli, P., Musina, R. On the existence of extremal functions for a weighted Sobolev embedding with critical exponent. Calc Var 8, 365–387 (1999). https://doi.org/10.1007/s005260050130
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DOI: https://doi.org/10.1007/s005260050130