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Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system
Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian
| 1. | Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Av. Diagonal 647. 08028 Barcelona |
| 2. | Centro de Matemática, Universidade de Coimbra, Apartado 3008, 3001–454 Coimbra, Portugal |
Under some assumptions on $f$ that make its growth comparable to $u^m$, we prove that every semi-stable solution is bounded if $m < m_{c s}$. Here, $m_{c s}=m_{c s}(N,p)$ is an explicit exponent which is optimal for the boundedness of semi-stable solutions. In particular, it is bigger than the critical Sobolev exponent $p^\star-1$.
We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal $L^\infty$ estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and $1 < p < 2$.
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