One dimensional symmetry of solutions to some anisotropic quasilinear elliptic equations in the plane
Giuseppe Riey Berardino Sciunzi
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.
Low dimensional instability for semilinear and quasilinear problems in $\mathbb{R}^N$
Daniele Castorina Pierpaolo Esposito Berardino Sciunzi
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

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