Universal estimates and Liouville theorems for superlinear problems without scale invariance

Abstract

<p style='text-indent:20px;'>We revisit rescaling methods for nonlinear elliptic and parabolic problems and show that, by suitable modifications, they may be used for nonlinearities that are not scale-invariant even asymptotically and whose behavior can be quite far from power like.</p><p style='text-indent:20px;'>In this enlarged framework, by adapting the doubling-rescaling method from [<xref ref-type="bibr" rid="b37">37</xref>, <xref ref-type="bibr" rid="b38">38</xref>], we show that the equivalence found there between universal estimates and Liouville theorems remains valid. In the parabolic case we also prove a Liouville type theorem for a rather large class of non scale-invariant nonlinearities. This leads to a number of new results for non scale-invariant elliptic and parabolic problems, concerning space or space-time singularity estimates, initial and final blow-up rates, universal and a priori bounds for global solutions, and decay rates in space and/or time.</p><p style='text-indent:20px;'>We illustrate our approach by a number of examples, which in turn give indication about the optimality of the estimates and of the assumptions.</p>