CPAA
One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane
Giuseppe Riey Berardino Sciunzi
Communications on Pure & Applied Analysis 2012, 11(3): 1157-1166 doi: 10.3934/cpaa.2012.11.1157
We prove one-dimensional symmetry of monotone solutions for some anisotropic quasilinear elliptic equations in the plane.
keywords: geometric analysis Degenerate anisotropic elliptic PDEs rigidity and symmetry results.
CPAA
Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$
Daniele Castorina Pierpaolo Esposito Berardino Sciunzi
Communications on Pure & Applied Analysis 2009, 8(6): 1779-1793 doi: 10.3934/cpaa.2009.8.1779
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
DCDS
Second order regularity for degenerate nonlinear elliptic equations
Annamaria Canino Elisa De Giorgio Berardino Sciunzi
Discrete & Continuous Dynamical Systems - A 2018, 38(8): 4231-4242 doi: 10.3934/dcds.2018184

We investigate the second order regularity of solutions to degenerate nonlinear elliptic equations.

keywords: Nonlinear elliptic equations regularity theory

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