# American Institute of Mathematical Sciences

March  2007, 6(1): 43-67. doi: 10.3934/cpaa.2007.6.43

## 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

Received  April 2006 Revised  July 2006 Published  December 2006

We consider nonnegative solutions of $-\Delta_p u=f(x,u)$, where $p>1$ and $\Delta_p$ is the $p$-Laplace operator, in a smooth bounded domain of $\mathbb R^N$ with zero Dirichlet boundary conditions. We introduce the notion of semi-stability for a solution (perhaps unbounded). We prove that certain minimizers, or one-sided minimizers, of the energy are semi-stable, and study the properties of this class of solutions.
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 $1 < p < 2$.
Citation: Xavier Cabré, Manel Sanchón. Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian. Communications on Pure & Applied Analysis, 2007, 6 (1) : 43-67. doi: 10.3934/cpaa.2007.6.43
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