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Mónica Clapp

Publications and source records attributed to Mónica Clapp.

At least 19 recordsLinked to original sources
arXiv2026

Sign-changing solutions to the Yamabe problem on a spherical cap

Spherical caps play a crucial role in establishing a criterion for the existence of solutions to the Yamabe problem on a compact Riemannian manifold with boundary, similar to the role played by the standard sphere in the problem on a closed Riemannian manifold. This problem is expressed in terms of a nonlinear boundary-value problem, where both the nonlinearity and the boundary condition are critical in the Sobolev sense. This work focuses on the existence of multiple solutions to the Yamabe problem on spherical caps. We show that if the spherical cap is contained in a hemisphere of the standard $n$-sphere and $n = 5$ or $n \geq 7$, the Yamabe problem has infinitely many sign-changing solutions. Our approach takes advantage of symmetries and is based on a careful analysis of the loss of compactness of the variational problem.

math.AP↗
arXiv2026

On the asymptotically linear problem for an elliptic equation with an indefinite nonlinearity

We study the semilinear elliptic problem \[ -Δu = Q_Ω |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where \( Q_Ω = χ_Ω - χ_{\mathbb{R}^N \setminus Ω} \) for a bounded smooth domain \( Ω\subset \mathbb{R}^N \), \( N \ge 3 \), and \( 1 < p < 2^{*} \). This equation arises in the study of optical waveguides and exhibits indefinite nonlinearity due to the sign-changing weight \( Q_Ω \). We prove that, for \( p > 2 \) sufficiently close to \( 2 \), the problem admits a unique positive solution, which is nondegenerate. Our approach combines a detailed analysis of an associated eigenvalue problem involving \( Q_Ω \) with variational methods and blow-up techniques in the asymptotically linear regime. We also provide a comprehensive study of the spectral properties of the corresponding linear problem, including the existence and qualitative behavior of eigenfunctions, sharp decay estimates, and symmetry results. In particular, we establish analogues of the Faber--Krahn and Hong--Krahn--Szeg{ö} inequalities in this non-standard setting.

math.AP↗
arXiv2026

Sign-changing solutions to the Yamabe problem on manifolds with boundary

Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is equivalent, in analytic terms, to finding a positive solution to a nonlinear boundary-value problem with critical growth. While the existence of positive solutions to this problem is by now well understood, the existence of sign-changing (nodal) solutions remains largely open. In this work we establish the existence of least-energy sign-changing solutions when the manifold is positive and the mean curvature of the boundary is a non-negative constant. More precisely, we prove that if $n\ge7$ and $M$ has a nonumbilic boundary point, then the problem admits least-energy nodal solutions. Our approach is variational and relies on the analysis of suitable conformal invariants and sharp energy estimates.

math.DG↗
arXiv2026

Sublinear elliptic equations with a sharp change of sign in the nonlinearity

We study the semilinear indefinite elliptic problem \[ -Δu = Q_Ω|u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_Ω= χ_Ω- χ_{\mathbb{R}^N \setminus Ω}$, $Ω\subset \mathbb{R}^N$ is a bounded smooth subset, $N \geq 3$, and $1 \leq p < 2$, with $p=1$ corresponding to the sign nonlinearity. Using a variational approach, we investigate the uniqueness or multiplicity of nonnegative solutions depending on the shape of $Ω$ and the existence of different types of nodal solutions. We also show that all solutions have compact support and analyze how the support of the ground state depends on $p$, proving convergence to the whole space as $p\to 2^{-}$ and identifying some qualitative features such as starshapedness and Lipschitz regularity of the support. We also establish a link between these problems and a two-phase Serrin-type torsion overdetermined problem.

math.AP↗
arXiv2025

Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity

We consider a nonautonomous semilinear elliptic problem where the power nonlinearity is multiplied by a discontinuous coefficient that equals one inside a bounded open set $Ω$ and it equals minus one in its complement. In the slightly subcritical regime, we prove the existence of concentrating positive and nodal solutions. Moreover, depending on the geometry of $Ω$, we establish multiplicity of positive solutions. Finally, in the critical case, we show the existence of a blow-up positive solution when $Ω$ has nontrivial topology. Our proofs rely on a Lyapunov-Schmidt reduction strategy which in these problems turns out to be remarkably simple. We take this opportunity to highlight certain aspects of the method that are often overlooked and present it in a more accessible and detailed manner for nonexperts.

math.AP↗
arXiv2025

Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity

We consider the nonlinear elliptic equation \begin{equation*} -Δu + V(x)u = f(u), \qquad u\in D^{1,2}_0(Ω), \end{equation*} in an exterior domain $Ω$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at infinity and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. Under weak symmetry assumptions, we provide conditions that guarantee that this problem has a prescribed number of sign-changing solutions. In particular, we show that in dimensions $N\geq 4$ there are numerous examples of exterior domains with finite symmetries in which the problem has a predetermined number of nodal solutions.

math.AP↗
arXiv2025

A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core

We consider the equation $$-Δu+u=Q_\varepsilon(x)|u|^{p-2}u,\qquad u\in H^1(\mathbb{R}^N),$$ where $Q_\varepsilon$ takes the value $1$ on each ball $B_\varepsilon(y)$, $y\in\mathbb{Z}^N$, and the value $-1$ elsewhere. We establish the existence of a least energy solution for each $\varepsilon\in(0,\frac{1}{2})$ and show that their $H^1$ and $L^p$ norms concentrate locally at points of $\mathbb{Z}^N$ as $\varepsilon\to 0$.

math.AP↗
arXiv2025

Critical equations with a sharp change of sign in the nonlinearity

We establish the existence and nonexistence of entire solutions to a semilinear elliptic problem whose nonlinearity is the critical power multiplied by a function that takes the value 1 in an open bounded region and the value -1 in its complement. The existence or not of solutions depends on the geometry of the bounded region, in a way analogous to what happens with the classical critical Dirichlet problem in a bounded domain. Our methods are variational and include the use of topological tools.

math.AP↗
arXiv2025

Entire solutions to a quasilinear purely critical competitive system

We establish the existence of a fully nontrivial solution with nonnegative components for a weakly coupled competitive system for the $p$-Laplacian in $\mathbb{R}^N$ whose nonlinear terms are purely critical. We also show that the purely critical equation for the $p$-Laplacian in $\mathbb{R}^N$ has infinitely many nodal solutions.

math.AP↗
arXiv2024

Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction

We study the existence and concentration of positive and nodal solutions to a Schrödinger equation in the presence of a shrinking self-focusing core of arbitrary shape. Via a suitable rescaling, the concentration gives rise to a limiting profile that solves a nonautonomous elliptic semilinear equation with a sharp sign change in the nonlinearity. We characterize the (radial or foliated Schwarz) symmetries and the (polynomial) decay of the least-energy positive and nodal limiting profiles.

math.AP↗
arXiv2024

On a Schrödinger system with shrinking regions of attraction

In this paper we consider a competitive weakly coupled elliptic system in which each species is attracted to a small region and repelled from its complement. In this setting, we establish the existence of infinitely many solutions and of a nonnegative least energy solution. We show that, as the regions of attraction shrink, least energy solutions of the system concentrate. We study this behavior and characterize their limit profile. In particular, we show that if each component of a least energy solution is attracted to a different region, then the components decouple in the limit, whereas if all the components are attracted to the same region, they remain coupled.

math.AP↗
arXiv2024

Optimal pinwheel partitions for the Yamabe equation

We establish the existence of an optimal partition for the Yamabe equation in the whole space made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to minus infinity, the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation that are different from those previously found in the by W.Y. Ding, and del Pino, Musso, Pacard and Pistoia

math.AP↗
arXiv2024

Optimal pinwheel partitions and pinwheel solutions to a nonlinear Schrödinger system

We establish the existence of a solution to a nonlinear competitive Schrödinger system whose scalar potential tends to a positive constant at infinity with an appropriate rate. This solution has the property that all components are invariant under the action of a group of linear isometries and each component is obtained from the previous one by composing it with some fixed linear isometry. We call it a pinwheel solution. We describe the asymptotic behavior of the least energy pinwheel solutions when the competing parameter tends to zero and to minus infinity. In the latter case the components are segregated and give rise to an optimal pinwheel partition for the Schrödinger equation, that is, a partition formed by invariant sets that are mutually isometric through a fixed isometry.

math.AP↗
arXiv2023

Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative components increases with the number of equations. Our proofs use a combination of four key elements: a convenient variational approach, the asymptotic behavior of solutions (concentration), the Lusternik-Schnirelman theory, and new estimates on the category of suitable configuration spaces.

math.AP↗
arXiv2023

Exponential decay of the solutions to nonlinear Schrödinger systems

We show that the components of finite energy solutions to general nonlinear Schrödinger systems have exponential decay at infinity. Our results apply to positive or sign-changing components, and to cooperative, competitive, or mixed-interaction systems. As an application, we use the exponential decay to derive an upper bound for the least possible energy of a solution with a prescribed number of positive and nonradial sign-changing components.

math.AP↗
arXiv2023

Pinwheel solutions to Schrödinger systems

We establish the existence of positive segregated solutions for competitive nonlinear Schrödinger systems in the presence of an external trapping potential, which have the property that each component is obtained from the previous one by a rotation, and we study their behavior as the forces of interaction become very small or very large. As a consequence, we obtain optimal partitions for the Schrödinger equation by sets that are linearly isometric to each other.

math.AP↗
arXiv2022

An upper bound for the least energy of a sign-changing solution to a zero mass problem

We give an upper bound for the least energy of a sign-changing solution to the the nonlinear scalar field equation $$-Δu = f(u), \qquad u\in D^{1,2}(\mathbb{R}^{N}),$$ where $N\geq5$ and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that $12c_0$ if $N=5,6$ and smaller than $10c_0$ if $N\geq 7$, where $c_0$ is the ground state energy.

math.AP↗