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Abstract
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 $1 < p < 2$.
Mathematics Subject Classification: $p$-Laplacian, semi-stable and extremal solutions, regularity.
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