arXiv subjects

Monica Clapp

Publications and source records attributed to Monica Clapp.

6 recordsLinked to original sources

arXiv2021

Fully nontrivial solutions to elliptic systems with mixed couplings

We study the existence of fully nontrivial solutions to the system $$-Δu_i+ λ_iu_i = \sum\limits_{j=1}^\ell β_{ij}|u_j|^p|u_i|^{p-2}u_i\ \hbox{in}\ Ω, \qquad i=1,\ldots,\ell,$$ in a bounded or unbounded domain $Ω$ in $\mathbb R^N,$ $N\ge 3$. The $λ_i$'s are real numbers, and the nonlinear term may have subcritical ($1 \frac{N}{N-2}$). The matrix $(β_{ij})$ is symmetric and admits a block decomposition such that the diagonal entries $β_{ii}$ are positive, the interaction forces within each block are attractive (i.e., all entries $β_{ij}$ in each block are non-negative) and the interaction forces between different blocks are repulsive (i.e., all other entries are non-positive). We obtain new existence and multiplicity results of fully nontrivial solutions, i.e., solutions where every component $u_i$ is nontrivial. We also find fully synchronized solutions (i.e., $u_i=c_i u_1$ for all $i=2,\ldots,\ell$) in the purely cooperative case whenever $p\in(1,2).$

math.AP↗

arXiv2020

A solution to a slightly subcritical elliptic problem with non-power nonlinearity

We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function. This is the first existence result for this type of generalized slightly subcritical problems.

math.AP↗

arXiv2017

Positive and nodal single-layered solutions to supercritical elliptic problems above the higher critical exponents

We study the problem% \[ -Δv+λv=| v| ^{p-2}v\text{ in }Ω,\text{\qquad}v=0\text{ on $\partialΩ$},\text{ }% \] for $λ\in\mathbb{R}$ and supercritical exponents $p,$ in domains of the form% \[ Ω:=\{(y,z)\in\mathbb{R}^{N-m-1}\times\mathbb{R}^{m+1}:(y,| z| )\inΘ\}, \] where $m\geq1,$ $N-m\geq3,$ and $Θ$ is a bounded domain in $\mathbb{R}% ^{N-m}$ whose closure is contained in $\mathbb{R}^{N-m-1}\times(0,\infty)$. Under some symmetry assumptions on $Θ$, we show that this problem has infinitely many solutions for every $λ$ in an interval which contains $[0,\infty)$ and $p>2$ up to some number which is larger than the $(m+1)^{st}$ critical exponent $2_{N,m}^{\ast}:=\frac{2(N-m)}{N-m-2}$. We also exhibit domains with a shrinking hole, in which there are a positive and a nodal solution which concentrate on a sphere, developing a single layer that blows up at an $m$-dimensional sphere contained in the boundary of $Ω,$ as the hole shrinks and $p\rightarrow2_{N,m}^{\ast}$ from above. The limit profile of the positive solution, in the transversal direction to the sphere of concentration, is a rescaling of the standard bubble, whereas that of the nodal solution is a rescaling of a nonradial sign-changing solution to the problem% \[ -Δu=| u| ^{2_{n}^{\ast}-2}u,\text{\qquad}u\in D^{1,2}(\mathbb{R}^{n}), \] where $2_{n}^{\ast}:=\frac{2n}{n-2}$ is the critical exponent in dimension $n.$\medskip

math.AP↗

arXiv2014

Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold

Given a smooth Riemannian manifold (M,g)we investigate the existence of positive solutions to a singularly perturbed supercritical elliptic equation which concentrate at some submanifold of M. We obtain a posive answer for some manifolds, which include warped products.

math.AP↗

arXiv2014

Semiclassical states for a static supercritical Klein-Gordon-Maxwell-Proca system on a closed Riemannian manifold

We establish the existence of semiclassical states for a nonlinear Klein-Gordon-Maxwell-Proca system in static form, with Proca mass 1 on a closed Riemannian manifold. Our results include manifolds of arbitrary dimension and allow supercritical nonlinearities. In particular, we exhibit a large class of 3-dimensional manifolds on which the system has semiclassical solutions for every exponent p>2. The solutions we obtain concentrate at closed submanifolds of positive dimension as the singular perturbation parameter goes to zero

math.AP↗

arXiv2012

Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains

Let $Γ$ denote a smooth simple curve in $\mathbb{R}^{N}$, $N\geq2$, possibly with boundary. Let $Ω_{R}$ be the open normal tubular neighborhood of radius 1 of the expanded curve $RΓ:=\{Rx\mid x\in Γ\smallsetminus\partialΓ\}$. Consider the superlinear problem $-Δu+λu=f(u)$ on the domains $Ω_{R}$, as $R\rightarrow \infty$, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along $RΓ$ with alternating signs. The function $f$ is superlinear at 0 and at $\infty$, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.

math.AP↗