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Mónica Clapp

Publications and source records attributed to Mónica Clapp.

50 records · Page 3Linked to original sources
arXiv2017

Multiplicity of nodal solutions to the Yamabe problem

Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-div_{g}(a\nabla u)+bu=c|u|^{2^{\ast}-2}u\quad on\ M$$ where $a,b,c\in C^{\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\nabla)+b$ is coercive, and $2^{\ast}=\frac{2m}{m-2}$ is the critical Sobolev exponent. In particular, if $R_{g}$ denotes the scalar curvature of $(M,g)$, we give conditions which guarantee that the Yamabe problem $$Δ_{g}u+\frac{m-2}{4(m-1} R_{g}u=κu^{2^{\ast}-2}\quad on\ M$$ admits a prescribed number of nodal solutions.

math.AP↗
arXiv2015

Symmetries, Hopf fibrations and supercritical elliptic problems

We consider the semilinear elliptic boundary value problem \[ -Δu=\left\vert u\right\vert ^{p-2}u\text{ in }Ω,\text{\quad }u=0\text{ on }\partialΩ, \] in a bounded smooth domain $Ω$ of $\mathbb{R}^{N}$ for supercritical exponents $p>\frac{2N}{N-2}.$ Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents.

math.AP↗
arXiv2015

Ground states of critical and supercritical problems of Brezis-Nirenberg type

We study the existence of symmetric ground states to the supercritical problem \[ -Δv=λv+\left\vert v\right\vert ^{p-2}v\text{ \ in }Ω,\qquad v=0\text{ on }\partialΩ, \] in a domain of the form \[ Ω=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert y\right\vert ,z\right) \inΘ\}, \] where $Θ$ is a bounded smooth domain such that $\overlineΘ \subset\left( 0,\infty\right) \times\mathbb{R}^{N-k-1},$ $1\leq k\leq N-3,$ $λ\in\mathbb{R},$ and $p=\frac{2(N-k)}{N-k-2}$ is the $(k+1)$-st critical exponent. We show that symmetric ground states exist for $λ$ in some interval to the left of each symmetric eigenvalue, and that no symmetric ground states exist in some interval $(-\infty,λ_{\ast})$ with $λ_{\ast}>0$ if $k\geq2.$ Related to this question is the existence of ground states to the anisotropic critical problem \[ -\text{div}(a(x)\nabla u)=λb(x)u+c(x)\left\vert u\right\vert ^{2^{\ast }-2}u\quad\text{in}\ Θ,\qquad u=0\quad\text{on}\ \partialΘ, \] where $a,b,c$ are positive continuous functions on $\overlineΘ.$ We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of $λ,$ and obtain a bifurcation result for ground states.

math.AP↗
arXiv2013

Positive solutions to a supercritical elliptic problem which concentrate along a thin spherical hole

We consider the supercritical problem \[ -Δv=|v|^{p-2}v in Θ_ε, v=0 on \partialΘ_ε, \] where $Θ$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $p>2^{\ast}:=2N/(N-2)$, and $Θ_ε$ is obtained by deleting the $ε$-neighborhood of some sphere which is embedded in $Θ$. In some particular situations we show that, for $ε>0$ small enough, this problem has a positive solution $v_ε$ and that these solutions concentrate and blow up along the sphere as $ε$ tends to 0. Our approach is to reduce this problem to a critical problem of the form \[ -Δu=Q(x)|u|^{4/(n-2)}u in Ω_ε, u=0 on \partialΩ_ε, \] in a punctured domain $Ω_ε:=\{x\inΩ:|x-ξ_{0}|>ε\}$ of lower dimension, by means of some Hopf map. We show that, if $Ω$ is a bounded smooth domain in $\mathbb{R}^{n}$, $n\geq3$, $ξ_{0} is inΩ,$ $Q is in C^{2}(\b{\Oarmega})$ is positive and $\nabla Q(ξ_{0})\neq0$ then, for $ε>0$ small enough, this problem has a positive solution $u_ε$, and that these solutions concentrate and blow up at $ξ_{0}$ as $ε$ goes to 0.

math.AP↗
arXiv2013

A supercritical elliptic problem in a cylindrical shell

We consider the problem \[ -Δu=|u|^{p-2}u in Ω, u=0 on \partialΩ, \] where $Ω:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}: 0<a<|y|<b<\infty\}$, $0\leq m\leq N-1$ and $N\geq2$. Let $2_{N,m}^{\ast}:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\ast}:=\infty$ if $m=N-2$ or $N-1$. We show that $2_{N,m}^{\ast}$ is the true critical exponent for this problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\ast}$ but there are no such solutions if $p\geq2_{N,m}^{\ast}$.

math.AP↗
arXiv2013

Multiplicity and asymptotic profile of 2-nodal solutions to a semilinear elliptic problem on a Riemannian manifold

We establish a lower bound for the number of sign changing solutions with precisely two nodal domains to the singularly perturbed nonlinear elliptic equation -ε^{2}Δ_{g}u+u=|u|^{p-2}u on an n-dimensional Riemannian manifold M, p in (2,2n/(n-2)), in terms of the cup-length of the configuration space of M. We give a precise description of the asymptotic profile of these solutions as ε goes to 0. We also provide new estimates for the cup-length of the configuration space of M.

math.AP↗
arXiv2012

Multiple solutions to nonlinear Schrödinger equations with singular electromagnetic potential

We consider the semilinear electromagnetic Schrödinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(Ω,\mathbb{C}), where $Ω=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq m\leq N$, $N\geq3$, $2^{\ast}:= 2N/(N-2)$ is the critical Sobolev exponent, $V$ is a Hardy term and $A$ is a singular magnetic potential of a particular form which includes the Aharonov-Bohm potentials. Under some symmetry assumptions on $A$ we obtain multiplicity of solutions satisfying certain symmetry properties.

math.AP↗
arXiv2012

Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem -Δu = |u|^{p-2}u in Ω, u=0 on \partialΩ, where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $Ω$ has nontrivial homology this problem has a positive solution for $p=2^{*}.$ However, this is not enough to guarantee existence in the supercritical case. For $p\geq 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show that for $p> 2(N-k)/(N-k-2)$ with $1\leq k\leq N-3$ there are bounded smooth domains in $\mathbb{R}^{N}$ whose cup-length is $k+1$ in which this problem does not have a nontrivial solution. For $N=4,8,16$ we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.

math.AP↗
arXiv2012

Positive and sign changing solutions to a nonlinear Choquard equation

We consider the problem \[-Δu + W(x)u = ((1/{|x|^α} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(Ω)\], where $Ω$ is an exterior domain in $\mathbb{R}^{N}$, $N\geq3,$ $α\in(0,N)$, $p\in[2,(2N-α)/(N-2)$, $W$ is continuous, $\inf_{\mathbb{R}^{N}}W>0,$ and $W(x)$ tends to a positive constant as $|x|$ tends to infinity. Under symmetry assumptions on $Ω$ and $W$, which allow finite symmetries, and some assumptions on the decay of $W$ at infinity, we establish the existence of a positive solution and multiple sign changing solutions to this problem, having small energy.

math.AP↗
arXiv2012

Boundary clustered layers near the higher critical exponents

We consider the supercritical problem {equation*} -Δu=|u| ^{p-2}u\text{\in}Ω,\quad u=0\text{\on}\partialΩ, {equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N}$ and $p$ smaller than the critical exponent $2_{N,k}^{\ast}:=\frac{2(N-k)}{N-k-2}$ for the Sobolev embedding of $H^{1}(\mathbb{R}^{N-k})$ in $L^{q}(\mathbb{R}^{N-k})$, $1\leq k\leq N-3.$ We show that in some suitable domains $Ω$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\partialΩ$ as $p$ approaches $2_{N,k}^{\ast}$ from below. Key words:Nonlinear elliptic boundary value problem; critical and supercritical exponents; existence of positive and sign changing solutions.

math.AP↗
arXiv2011

Intertwining semiclassical solutions to a Schrödinger-Newton system

We study the problem (-ε\mathrm{i}\nabla+A(x)) ^{2}u+V(x)u=ε^{-2}(\frac{1}{|x|}\ast|u|^{2}) u, u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),\text{\ \ \ \}ε\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magnetic potential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exterior electric potential, and $ε$ is a small positive number. If A=0 and $ε=\hbar$ is Planck's constant this problem is equivalent to the Schrödinger-Newton equations proposed by Penrose in \cite{pe2}\ to describe his view that quantum state reduction occurs due to some gravitational effect. We assume that $A$ and $V$ are compatible with the action of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for any given homomorphism $τ:G\rightarrow\mathbb{S}^{1}$ into the unit complex numbers, we show that there is a combined effect of the symmetries and the potential $V$ on the number of semiclassical solutions $u:\mathbb{R}% ^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=τ(g)u(x)$ for all $g\in G$, $x\in\mathbb{R}^{3}$. We also study the concentration behavior of these solutions as $ε\rightarrow0.\medskip$

math.AP↗
arXiv2011

Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation [(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^α}\ast |u|^{p}) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}%] where $A\ $is a real valued vector potential, $V$ is a real valued scalar potential$,$ $N\geq3$, $α\in(0,N)$ and $2-(α/N) <p<(2N-α)/(N-2)$. \ We assume that both $A$ and $V$ are compatible with the action of some group $G$ of linear isometries of $\mathbb{R}^{N}$. We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition \[ u(gx)=τ(g)u(x)\text{\ \ \ for all}g\in G,\text{}x\in\mathbb{R}^{N}, \] where $τ:G\rightarrow\mathbb{S}^{1}$ is a given group homomorphism into the unit complex numbers.

math.AP↗
arXiv2007

Singular limits for the bi-laplacian operator with exponential nonlinearity in $\R^4$

Let $Ω$ be a bounded smooth domain in $\mathbb{R}^{4}$ such that for some integer $d\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem {Δ^{2}u-ρ^{4}k(x)e^{u}=0 & \hbox{in}Ω, u=Δu=0 & \hbox{on}\partialΩ, has a solution blowing-up, as $ρ\to0$, at $m$ points of $Ω$, for any given number $m$.

math.AP↗