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A strong comparison principle for the -Laplacian

Authors:	Paolo Roselli Berardino Sciunzi

Institution:	Dipartimento di Matematica, Universà di Roma ``Tor Vergata', Via della Ricerca Scientifica 00133 Roma, Italy ; Dipartimento di Matematica, Università di Roma ``Tor Vergata', Via della Ricerca Scientifica, 00133 Roma, Italy

Abstract:	We consider weak solutions of the differential inequality of p-Laplacian type $\displaystyle - \Delta_p u - f(u) \le - \Delta_p v - f(v)$ such that $u\leq v$ on a smooth bounded domain in $\mathbb{R}^N$ and either or is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that on the boundary of the domain we prove that , and assuming that $u\equiv v\equiv0$ on the boundary of the domain we prove unless $u \equiv v$ . The novelty is that the nonlinearity is allowed to change sign. In particular, the result holds for the model nonlinearity $f(s) = s^q - \lambda s^{p-1}$ with .

Keywords:	$p$-Laplace operator geometric and qualitative properties of the solutions comparison principle

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