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November  2009, 8(6): 1779-1793. doi: 10.3934/cpaa.2009.8.1779

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

Daniele Castorina 1, , Pierpaolo Esposito 2, and  Berardino Sciunzi 3,

1. 

Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133 Roma, Italy
2. 

Dipartimento di Matematica, Universitá di Roma Tre, Largo San Leonardo Murialdo, 1, I-00146 Roma, Italy
3. 

Dipartimento di Matematica, Universitá della Calabria, V. P. Bucci, I-87036 Arcavacata di Rende (CS), Italy

Received  June 2008 Revised  April 2009 Published  August 2009

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.
Keywords: critical dimension., p−Laplace operator, linear instability.
Mathematics Subject Classification: Primary: 35J60, 35J70; Secondary: 35B0.
Citation: Daniele Castorina, Pierpaolo Esposito, Berardino Sciunzi. Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1779-1793. doi: 10.3934/cpaa.2009.8.1779
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