arXiv subjects
Publications and source records attributed to A. Pistoia.
We construct sign-changing concentrating solutions for a mean field equation describing turbulent Euler flows with variable vortex intensities and arbitrary orientation. We study the effect of variable intensities and orientation on the bubbling profile and on the location of the vortex points.
We consider the critical heat equation \begin{equation} \label{CH}\tag{CH} \begin{array}{lr} v_t-Δv =|v|^{\frac{4}{n-2}}v & Ω_ε\times (0, +\infty) \\ v=0 & \partialΩ_ε\times (0, +\infty) \\ v=v_0 & \mbox{ in } Ω_ε\times \{t=0\} \end{array} \end{equation} in $Ω_ε:=Ω\setminus B_ε(x_0)$ where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N\geq 3$ and $B_ε(x_0)$ is a ball of $\mathbb R^N$ of center $x_0\inΩ$ and radius $ε>0$ small. \\ We show that if $ε>0$ is small enough, then there exists a sign-changing stationary solution $ϕ_ε$ of \eqref{CH} such that the solution of \eqref{CH} with initial value $v_0=λϕ_ε$ blows up in finite time if $|λ-1|>0$ is sufficiently small.\\ This shows in particular that the set of the initial conditions for which the solution of \eqref{CH} is global and bounded is not star-shaped.
We study the critical problem {equation} {{array}{ll} Δ^{2}u=u^{\frac{N+4}{N-4}} & {in}Ω\setminus \bar{B(ξ_0,\varepsilon)},\medskip u>0&{in}Ω\setminus \bar{B(ξ_0,\varepsilon)},\medskip u=Δu=0 & {on}\partial (Ω\setminus \bar{B(ξ_0,\varepsilon)}),{array}. \tag{P$_\varepsilon$} {equation} where $Ω$ is an open bounded domain in $\mathbb{R}^N$, $N\ge5$, $ξ_0\inΩ$ and $B(ξ_0,\varepsilon)$ is the ball centered at $ξ_0$ with radius $\varepsilon>0$ small enough. We construct solutions of (P$_\varepsilon$) blowing-up at the center of the hole as the size of the hole goes to zero.