Searcharxiv⌕ Search

arXiv subjects

Mónica Clapp

Publications and source records attributed to Mónica Clapp.

At least 37 records · Page 2Linked to original sources
arXiv2022

Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems

We study the existence and asymptotic behavior of solutions having positive and sign-changing components to the singularly perturbed system of elliptic equations \begin{equation*} \begin{cases} -\varepsilon^2Δu_i+u_i=μ_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ellλ_{ij}β_{ij}|u_j|^{α_{ij}}|u_i|^{β_{ij} -2}u_i,\\ u_i \in H^1_0(Ω), \quad u_i\neq 0, \qquad i=1,\ldots,\ell, \end{cases} \end{equation*} in a bounded domain $Ω$ in $\mathbb{R}^N$, with $N\geq 4$, $\varepsilon>0$, $μ_i>0$, $λ_{ij}=λ_{ji}<0$, $α_{ij}, β_{ij}>1$, $α_{ij}=β_{ji}$, $α_{ij} + β_{ij} = p\in (2,2^*)$, and $2^{*}:=\frac{2N}{N-2}$. If $Ω$ is the unit ball we obtain solutions with a prescribed combination of positive and nonradial sign-changing components exhibiting two different types of asymptotic behavior as $\varepsilon\to 0$: solutions whose limit profile is a rescaling of a solution with positive and nonradial sign-changing components of the limit system \begin{equation*} \begin{cases} -Δu_i+u_i=μ_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ellλ_{ij}β_{ij}|u_j|^{α_{ij}}|u_i|^{β_{ij} -2}u_i,\\ u_i \in H^1(\mathbb{R}^N), \quad u_i\neq 0, \qquad i=1,\ldots,\ell, \end{cases} \end{equation*} and solutions whose limit profile is a solution of the uncoupled system, i.e., after rescaling and translation, the limit profile of the $i$-th component is a positive or a nonradial sign-changing solution to the equation $$-Δu+u=μ_i|u|^{p-2}u,\qquad u \in H^1(\mathbb{R}^N), \qquad u\neq 0.$$

math.AP↗
arXiv2022

Energy estimates for seminodal solutions to an elliptic system with mixed couplings

We study the system of semilinear elliptic equations $$-Δu_i+ u_i = \sum_{j=1}^\ell β_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1(\mathbb{R}^N),\qquad i=1,\ldots,\ell,$$ where $N\geq 4$, $1<p<\frac{N}{N-2}$, and the matrix $(β_{ij})$ is symmetric and admits a block decomposition such that the entries within each block are positive or zero and all other entries are negative. We provide simple conditions on $(β_{ij})$, which guarantee the existence of fully nontrivial solutions, i.e., solutions all of whose components are nontrivial. We establish existence of fully nontrivial solutions to the system having a prescribed combination of positive and nonradial sign-changing components, and we give an upper bound for their energy when the system has at most two blocks. We derive the existence of solutions with positive and nonradial sign-changing components to the system of singularly perturbed elliptic equations $$-\varepsilon^2Δu_i+ u_i = \sum_{j=1}^\ell β_{ij}|u_j|^p|u_i|^{p-2}u_i, \qquad u_i\in H^1_0(B_1(0)),\qquad i=1,\ldots,\ell,$$ in the unit ball, exhibiting two different kinds of asymptotic behavior: solutions whose components decouple as $\varepsilon\to 0$, and solutions whose components remain coupled all the way up to their limit.

math.AP↗
arXiv2022

Normalized solutions to a non-variational Schrödinger system

We establish the existence of positive normalized (in the $L^2$ sense) solutions to non-variational weakly coupled elliptic systems of $\ell$ equations. We consider couplings of both cooperative and competitive type. We show the problem can be formulated as an operator equation on the product of $\ell$ $L^2$-spheres and apply a degree-theoretical argument on this product to obtain existence.

math.AP↗
arXiv2021

Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$. To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if $\dim M\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For $\ell=2$ the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

math.AP↗
arXiv2021

Critical polyharmonic systems and optimal partitions

We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in R^N which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to $-\infty$. 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 R^N. We give a detailed description of the shape of these domains.

math.AP↗
arXiv2021

Non-variational weakly coupled elliptic systems

We establish the existence of a nonnegative fully nontrivial solution to a non-variational weakly coupled competitive elliptic system. We show that this kind of solutions belong to a topological manifold of Nehari-type, and apply a degree-theoretical argument on this manifold to derive existence.

math.AP↗
arXiv2020

Solutions to indefinite weakly coupled cooperative elliptic systems

We study the elliptic system \begin{equation*} \begin{cases} -Δu_1 - κ_1u_1 = μ_1|u_1|^{p-2}u_1 + λα|u_1|^{α-2}|u_2|^βu_1, \\ -Δu_2 - κ_2u_2 = μ_2|u_2|^{p-2}u_2 + λβ|u_1|^α|u_2|^{β-2}u_2, \\ u_1,u_2\in D^{1,2}_0(Ω), \end{cases} \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N\geq 3$, $κ_1,κ_2\in\mathbb{R}$, $μ_1,μ_2,λ>0$, $α,β>1$, and $α+ β= p\le 2^*:=\frac{2N}{N-2}$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $λ$ sufficiently large. If $p=2^*$ and $κ_1,κ_2>0$ we establish the existence of a ground state for $λ$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $κ_1$ nor $κ_2$ are Dirichlet eigenvalues of $-Δ$ in $Ω$.

math.AP↗
arXiv2019

Phase separation, optimal partitions, and nodal solutions to the Yamabe equation on the sphere

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.

math.AP↗
arXiv2019

Entire nodal solutions to the critical Lane-Emden system

We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-Δu=|v|^{q-2}v,\quad -Δv=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical hyperbola $\frac{1}{p}+\frac{1}{q}=\frac{N-2}{N}$. These solutions are nonradial and arise as limit profiles of symmetric sign-changing minimizing sequences for a critical higher-order problem in a bounded domain.

math.AP↗
arXiv2019

A simple variational approach to weakly coupled competitive elliptic systems

The main purpose of this paper is to exhibit a simple variational setting for finding fully nontrivial solutions to the weakly coupled elliptic system (1.1). We show that such solutions correspond to critical points of a $\mathcal{C}^1$-functional $Ψ:\mathcal{U}\to\mathbb{R}$ defined in an open subset $\mathcal{U}$ of the product $\mathcal{T}:=S_1\times\cdots\times S_M$ of unit spheres $S_i$ in an appropriate Sobolev space. We use our abstract setting to extend and complement some known results for the system (1.1).

math.AP↗
arXiv2019

Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone

We study the critical Neumann problem \begin{equation*} \begin{cases} -Δu = |u|^{2^*-2}u &\text{in }Σ_ω,\\ \quad\frac{\partial u}{\partialν}=0 &\text{on }\partialΣ_ω, \end{cases} \end{equation*} in the unbounded cone $Σ_ω:=\{tx:x\inω\text{ and }t>0\}$, where $ω$ is an open connected subset of the unit sphere $\mathbb{S}^{N-1}$ in $\mathbb{R}^N$ with smooth boundary, $N\geq 3$ and $2^*:=\frac{2N}{N-2}$. We assume that some local convexity condition at the boundary of the cone is satisfied. If $ω$ is symmetric with respect to the north pole of $\mathbb{S}^{N-1}$, we establish the existence of a nonradial sign-changing solution. On the other hand, if the volume of the unitary bounded cone $Σ_ω\cap B_1(0)$ is large enough (but possibly smaller than half the volume of the unit ball $B_1(0)$ in $\mathbb{R}^N$), we establish the existence of a positive nonradial solution.

math.AP↗
arXiv2018

Multiple solutions to weakly coupled supercritical elliptic systems

We study a weakly coupled supercritical elliptic system of the form \begin{equation*} \begin{cases} -Δu = |x_2|^γ\left(μ_{1}|u|^{p-2}u+λα|u|^{α-2}|v|^βu \right) & \text{in }Ω,\\ -Δv = |x_2|^γ\left(μ_{2}|v|^{p-2}v+λβ|u|^α|v|^{β-2}v \right) & \text{in }Ω,\\ u=v=0 & \text{on }\partialΩ, \end{cases} \end{equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, $γ\geq 0$, $μ_{1},μ_{2}>0$, $λ\in\mathbb{R}$, $α, β>1$, $α+β= p$, and $p\geq 2^{*}:=\frac{2N}{N-2}$. We assume that $Ω$ is invariant under the action of a group $G$ of linear isometries, $\mathbb{R}^{N}$ is the sum $F\oplus F^\perp$ of $G$-invariant linear subspaces, and $x_2$ is the projection onto $F^\perp$ of the point $x\inΩ$. Then, under some assumptions on $Ω$ and $F$, we establish the existence of infinitely many fully nontrivial $G$-invariant solutions to this system for $p\geq 2^*$ up to some value which depends on the symmetries and on $γ$. Our results apply, in particular, to the system with pure power nonlinearity ($γ=0$), and yield new existence and multiplicity results for the supercritical Hénon-type equation $$-Δw = |x_2|^γ\,|w|^{p-2}w \quad\text{in }Ω, \qquad w=0 \quad\text{on }\partialΩ.$$

math.AP↗
arXiv2018

Positive multipeak solutions to a zero mass problem in exterior domains

We establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass $$-Δu = f(u), \qquad u\in D_0^{1,2}(Ω_R),$$ where $Ω_R:=\{x \in \mathbb{R}^N:|u|>R\}$ with $R>0$, $N\geq4$, and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. We show that the number of positive multipeak solutions becomes arbitrarily large as $R \to \infty$.

math.AP↗
arXiv2018

Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains

We study the weakly coupled critical elliptic system \begin{equation*} \begin{cases} -Δu=μ_{1}|u|^{2^{*}-2}u+λα|u|^{α-2}|v|^βu & \text{in }Ω,\\ -Δv=μ_{2}|v|^{2^{*}-2}v+λβ|u|^α|v|^{β-2}v & \text{in }Ω,\\ u=v=0 & \text{on }\partialΩ, \end{cases} \end{equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, $2^{*}:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $μ_{1},μ_{2}>0$, $α, β>1$, $α+β=2^{*}$ and $λ\in\mathbb{R}$. We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on $Ω$, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as $λ\to -\infty$. We also obtain existence of infinitely many solutions to this system in $Ω=\mathbb{R}^N$.

math.AP↗
arXiv2017

Existence and phase separation of entire solutions to a pure critical competitive elliptic system

We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \[ \left\{ \begin{tabular} [c]{l}% $-Δu=μ_{1}|u|^{{2}^{\ast}-2}u+λα|u|^{α-2}|v|^{β}u,$\\ $-Δv=μ_{2}|v|^{{2}^{\ast}-2}v+λβ|u|^α|v|^{β{-2}% }v,$\\ $u,v\in D^{1,2}(\mathbb{R}^{N}),$% \end{tabular} \ \right. \] where $N\geq4,$ $2^{\ast}:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $α,β\in(1,2],$ $α+β=2^{\ast},$ $μ_{1},μ_{2}>0,$ and $λ<0.$ We show that these solutions exhibit phase separation as $λ\rightarrow-\infty,$ and we give a precise description of their limit domains. If $μ_{1}=μ_{2}$ and $α=β$, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.

math.AP↗
arXiv2017

Existence of a positive solution to a nonlinear scalar field equation with zero mass at infinity

We establish the existence of a positive solution to the problem $$-Δu+V(x)u=f(u),\qquad u\in D^{1,2}(\mathbb{R}^{N}),$$ for $N\geq3$, when the nonlinearity $f$ is subcritical at infinity and supercritical near the origin, and the potential $V$ vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.

math.AP↗
arXiv2017

Entire nodal solutions to the pure critical exponent problem for the $p$-Laplacian

We establish the existence of multiple sign-changing solutions to the quasilinear critical problem $$-Δ_{p} u=|u|^{p^*-2}u, \qquad u\in D^{1,p}(\mathbb{R}^{N}),$$ for $N\geq4$, where $Δ_{p}u:=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplace operator, $1<p<N$ and $p^*:=\frac{Np}{N-p}$ is the critical Sobolev exponent

math.AP↗