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


Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, Av. Diagonal 647. 08028 Barcelona


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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