Seminario EDP UAM-ICMAT
Non-radial and segregated solutions for critical Choquard equations and systems
Ponente: Sabrina Caputo (Università degli Studi di Bari Aldo Moro)Fecha: viernes 30 de octubre de 2026 - 12:30Lugar: Aula Gris 1, ICMAT
Resumen:
In this work, we present recent results on the existence and qualitative behavior of solutions to elliptic equations and systems of Choquard type in the critical growth regime.
In the first part, we investigate a class of critical Choquard equations with a symmetric potential:
-\Delta u+V(|x|)u=(|x|^{-\mu}*|u|^{2^*_\mu})|u|^{2^*_\mu-2}u, \in mathbb{R}^N
where $V(|x|)$ is a bounded, nonnegative and symmetric potential in $\mathbb{R}^N$ with $N\geq 5$, $ 0<\mu\leq 4$ and $2^*_\mu:=\frac{2N-\mu}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
By applying a finite dimensional method we prove that if $r^2V(r)$ has a local maximum point or local minimum point at $r_0>0$ with $V(r_0)>0$ then the problem has infinitely many non-radial solutions with arbitrary large energies.
In the second part, the analysis is extended to a coupled elliptic system of critical Choquard equations in the presence of a small interspecies repulsive force:
-\Delta u_i=(|x|^{-\mu}*|u_i|^{4-\frac{\mu}{2}})|u_i|^{2-\frac{\mu}{2}}u_i+\sum_{j=1\, j\not =i}^2\beta u_iu_j^2\quad \hbox{in} \, \, \mathbb{R}^4,\quad i=1,2.
If $\beta<0, |\beta|$ is small enough, by utilizing perturbative and reduction techniques we build segregated solutions where the two components exhibit radically distinct asymptotic profiles: the first component looks like the radial positive solution of the single equation, while the second one blows-up at the $k$ vertices of a regular polygon.
Collectively, these results highlight the crucial interplay of geometric symmetries, external potentials and competitive interactions in dictating the multiplicity and concentration patterns for Hartree-type problems.
