arXiv subjects
Publications and source records attributed to Isabella Ianni.
We study the existence of nonradial sign-changing solutions to the Schroedinger-Poisson system in dimension N>=3. We construct nonradial sign-changing multi-peak solutions whose peaks are displaced in suitable symmetric configurations and collapse to the same point. The proof is based on the Lyapunov-Schmidt reduction.
We consider the semilinear Lane-Emden problem \begin{equation}\label{problemAbstract}\left\{ \begin{array}{lr} -Δu= |u|^{p-1}u\qquad \mbox{ in }Ω\\ u=0\qquad\qquad\qquad\mbox{ on }\partial Ω\end{array} \right.\tag{$\mathcal E_p$} \end{equation} where $p>1$ and $Ω$ is a smooth bounded symmetric domain of $\mathbb R^2$. We show that for families $(u_p)$ of sign-changing symmetric solutions of \eqref{problemAbstract} an upper bound on their Morse index implies concentration of the positive and negative part, $u_p^\pm$, at the same point, as $p\to+\infty$. Then an asymptotic analysis of $u_p^+$ and $u_p^-$ shows that the asymptotic profile of $(u_p)$, as $p\to+\infty$, is that of a tower of two different bubbles.
We consider the semilinear heat equation \begin{equation}\label{problemAbstract}\left\{\begin{array}{ll}v_t-Δv= |v|^{p-1}v & \mbox{in}Ω\times (0,T)\\ v=0 & \mbox{on}\partial Ω\times (0,T)\\ v(0)=v_0 & \mbox{in}Ω\end{array}\right.\tag{$\mathcal P_p$} \end{equation} where $p>1$, $Ω$ is a smooth bounded domain of $\mathbb R^2$, $T\in (0,+\infty]$ and $v_0$ belongs to a suitable space. We give general conditions for a family $u_p$ of sign-changing stationary solutions of \eqref{problemAbstract}, under which the solution of \eqref{problemAbstract} with initial value $v_0=λu_p$ blows up in finite time if $|λ-1|>0$ is sufficiently small and $p$ is sufficiently large. Since for $λ=1$ the solution is global, this shows that, in general, the set of the initial conditions for which the solution is global is not star-shaped with respect to the origin. In previous paper by Dickstein, Pacella and Sciunzi this phenomenon has already been observed in the case when the domain is a ball and the sign changing stationary solution is radially symmetric. Our conditions are more general and we provide examples of stationary solutions $u_p$ which are not radial and exhibit the same behavior.
We consider the semilinear Lane-Emden problem in a smooth bounded domain of the plane. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions as the exponent p of the nonlinearity goes to infinity. Among other results we show, under some symmetry assumptions on the domain, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p goes to infinity, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in the plane.
We prove the existence of a new type of solutions to a nonlinear Schrödinger system. These solutions, which we call "multi-speeds solitary waves", are behaving at large time as a couple of scalar solitary waves traveling at different speeds. The proof relies on the construction of approximations of the multi-speeds solitary waves by solving the system backwards in time and using energy methods to obtain uniform estimates.
We consider the semilinear Lane Emden problem in a smooth bounded simply connected domain in the plane, invariant by the action of a finite symmetry group G. We show that if the orbit of each point in the domain, under the action of the group G, has cardinality greater than or equal to four then, for p sufficiently large, there exists a sign changing solution of the problem with two nodal regions whose nodal line does not touch the boundary of the domain. This result is proved as a consequence of an analogous result for the associated parabolic problem.
We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a "Newtonian" nonlocal term.
We consider the Schrödinger-Poisson-Slater (SPS) system in $\R^3$ and a nonlocal SPS type equation in balls of $\mathbb R^3$ with Dirichlet boundary conditions. We show that for every $k\in\mathbb N$ each problem considered admits a nodal radially symmetric solution which changes sign exacly $k$ times in the radial variable. Moreover when the domain is the ball of $\mathbb R^3$ we obtain the existence of radial global solutions for the associated nonlocal parabolic problem having $k+1$ nodal regions at every time.
In this paper the following version of the Schrodinger-Poisson-Slater problem is studied: $$ - Δu + (u^2 \star \frac{1}{|4πx|}) u=μ|u|^{p-1}u, $$ where $u: \R^3 \to \R$ and $μ>0$. The case $p <2$ being already studied, we consider here $p \geq 2$. For $p>2$ we study both the existence of ground and bound states. It turns out that $p=2$ is critical in a certain sense, and will be studied separately. Finally, we prove that radial solutions satisfy a point-wise exponential decay at infinity for $p>2$.