scholarly journals Critical polyharmonic systems and optimal partitions

2021 ◽  
Vol 20 (11) ◽  
pp. 3991
Author(s):  
Mónica Clapp ◽  
Juan Carlos Fernández ◽  
Alberto Saldaña
Keyword(s):  
Existence Of Solutions ◽  
Optimal Partition ◽  
Partition Problem ◽  
Competitive System ◽  
Polyharmonic Equations ◽  
Optimal Partitions ◽  
Weakly Coupled ◽  
Coupling Parameters ◽  
Least Energy Solutions ◽  
Least Energy

<p style='text-indent:20px;'>We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to <inline-formula><tex-math id="M2">\begin{document}$ -\infty $\end{document}</tex-math></inline-formula>. We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^N $\end{document}</tex-math></inline-formula>. We give a detailed description of the shape of these domains.</p>

scholarly journals Phase Separation, Optimal Partitions, and Nodal Solutions to the Yamabe Equation on the Sphere

2020 ◽  
Author(s):  
Mónica Clapp ◽  
Alberto Saldaña ◽  
Andrzej Szulkin
Keyword(s):  
Optimal Partition ◽  
Partition Problem ◽  
Round Sphere ◽  
Nodal Solutions ◽  
Nodal Domains ◽  
Optimal Partitions ◽  
Yamabe Equation ◽  
Weakly Coupled ◽  
Least Energy Solutions ◽  
Least Energy

Abstract We study an optimal $M$-partition problem for the Yamabe equation on the round sphere, in the presence of some particular symmetries. We show that there is a correspondence between solutions to this problem and least energy sign-changing symmetric solutions to the Yamabe equation on the sphere with precisely $M$ nodal domains. The existence of an optimal partition is established through the study of the limit profiles of least energy solutions to a weakly coupled competitive elliptic system on the sphere.

scholarly journals Yamabe systems and optimal partitions on manifolds with symmetries

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mónica Clapp ◽  
Angela Pistoia
Keyword(s):  
Riemannian Manifold ◽  
Invariant Solution ◽  
Optimal Partition ◽  
Nodal Domains ◽  
Optimal Partitions ◽  
Yamabe Equation ◽  
Weakly Coupled ◽  
Group Of Isometries ◽  
Least Energy ◽  
Competition Parameter

<p style='text-indent:20px;'>We prove the existence of regular optimal <inline-formula><tex-math id="M1">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant partitions, with an arbitrary number <inline-formula><tex-math id="M2">\begin{document}$ \ell\geq 2 $\end{document}</tex-math></inline-formula> of components, for the Yamabe equation on a closed Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ (M,g) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math></inline-formula> is a compact group of isometries of <inline-formula><tex-math id="M5">\begin{document}$ M $\end{document}</tex-math></inline-formula> with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of <inline-formula><tex-math id="M6">\begin{document}$ \ell $\end{document}</tex-math></inline-formula> equations, related to the Yamabe equation. We show that this system has a least energy <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to <inline-formula><tex-math id="M8">\begin{document}$ -\infty $\end{document}</tex-math></inline-formula>, giving rise to an optimal partition. For <inline-formula><tex-math id="M9">\begin{document}$ \ell = 2 $\end{document}</tex-math></inline-formula> the optimal partition obtained yields a least energy sign-changing <inline-formula><tex-math id="M10">\begin{document}$ G $\end{document}</tex-math></inline-formula>-invariant solution to the Yamabe equation with precisely two nodal domains.</p>

Least Energy Solutions with Sign Information for Parametric Double Phase Problems

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro ◽  
Francesca Vetro
Keyword(s):  
Double Phase ◽  
Least Energy Solutions ◽  
Least Energy

scholarly journals The asymptotically linear Hénon problem

2020 ◽  
pp. 2050042
Author(s):  
Anna Lisa Amadori
Keyword(s):  
Boundary Conditions ◽  
Morse Index ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Radial Solutions ◽  
Asymptotically Linear ◽  
Planar Case ◽  
Asymptotic Profile ◽  
Least Energy Solutions ◽  
Least Energy

In this paper, we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent [Formula: see text] is close to [Formula: see text]. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle [Formula: see text]. By considerations based on the Morse index we see that, depending on the values of [Formula: see text] and [Formula: see text], such least energy solutions can be radial, or nonradial and different one from another.

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