Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$

Daniele Castorina; Pierpaolo  Esposito; Berardino Sciunzi

doi:10.3934/cpaa.2009.8.1779

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2009, Volume 8, Issue 6: 1779-1793. Doi: 10.3934/cpaa.2009.8.1779

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Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$

1.
Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma
2.
Dipartimento di Matematica, Universitá di Roma Tre, Largo San Leonardo Murialdo, 1, I-00146 Roma
3.
Dipartimento di Matematica, Universitá della Calabria, V. P. Bucci, I-87036 Arcavacata di Rende (CS)

Received: June 2008

Revised: April 2009

Published: November 2009

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Abstract

Stability properties for solutions of $-\Delta_m(u)=f(u)$ in $\mathbb{R}^N$ are investigated, where $N\geq 2$ and $m \geq 2$. The aim is to identify a critical dimension $N^\#$ so that every non-constant solution is linearly unstable whenever $2\leq N < N^\#$. For positive, increasing and convex nonlinearities $f(u)$, global bounds on $\frac{f \, f''}{(f')^2}$ allows us to find a dimension $N^\#$, which is optimal for exponential and power nonlinearities. In the radial setting we can deal more generally with $C^1-$nonlinearities and the dimension $N^\#$ we find is still optimal.

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Mathematics Subject Classification: Primary: 35J60, 35J70; Secondary: 35B05.

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Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$

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