Mónica Clapp
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View article: A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core
A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core Open
We consider the equation $$\begin{aligned} -\Delta u+u=Q_\varepsilon (x)|u|^{p-2}u,\quad u\in H^1(\mathbb {R}^N), \end{aligned}$$ where $$Q_\varepsilon $$ takes the value 1 on each ball $$B_\v…
View article: Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity
Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity Open
We consider the nonlinear elliptic equation \begin{equation*} -Δu + V(x)u = f(u), \qquad u\in D^{1,2}_0(Ω), \end{equation*} in an exterior domain $Ω$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at infinity and th…
View article: A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core
A concentration phenomenon for a semilinear Schrödinger equation with periodic self-focusing core Open
We consider the equation $$-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 exis…
View article: Critical equations with a sharp change of sign in the nonlinearity
Critical equations with a sharp change of sign in the nonlinearity Open
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 com…
View article: Entire solutions to a quasilinear purely critical competitive system
Entire solutions to a quasilinear purely critical competitive system Open
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 c…
View article: Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity
Multiple solutions to a semilinear elliptic equation with a sharp change of sign in the nonlinearity Open
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 subc…
View article: On a Schrödinger system with shrinking regions of attraction
On a Schrödinger system with shrinking regions of attraction Open
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…
View article: Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction
Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction Open
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 p…
View article: Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation
Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation Open
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 ellip…
View article: Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system
Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system Open
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…
View article: Optimal pinwheel partitions and pinwheel solutions to a nonlinear Schrödinger system
Optimal pinwheel partitions and pinwheel solutions to a nonlinear Schrödinger system Open
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 invaria…
View article: An upper bound for the least energy of a sign-changing solution to a zero mass problem
An upper bound for the least energy of a sign-changing solution to a zero mass problem Open
We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation − Δ u = f ( u ) , u ∈ D 1,2 ( R N ) , $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\righ…
View article: A strong unique continuation property for weakly coupled elliptic systems
A strong unique continuation property for weakly coupled elliptic systems Open
We establish the validity of a strong unique continuation property for weakly coupled elliptic systems, including competitive ones. Our proof exploits the system structure and uses Carleman estimates. We apply this result to obtain some no…
View article: Optimal pinwheel partitions for the Yamabe equation
Optimal pinwheel partitions for the Yamabe equation Open
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 th…
View article: Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system
Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system Open
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…
View article: Exponential decay of the solutions to nonlinear Schrödinger systems
Exponential decay of the solutions to nonlinear Schrödinger systems Open
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-intera…
View article: Normalized solutions to a non-variational Schrödinger system
Normalized solutions to a non-variational Schrödinger system Open
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 ca…
View article: Exponential decay of the solutions to nonlinear Schrödinger systems
Exponential decay of the solutions to nonlinear Schrödinger systems Open
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-intera…
View article: Pinwheel solutions to Schrödinger systems
Pinwheel solutions to Schrödinger systems Open
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…
View article: An upper bound for the least energy of a sign-changing solution to a zero mass problem
An upper bound for the least energy of a sign-changing solution to a zero mass problem Open
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}),$$ where $N\geq5$ and the nonlinearity $f$ is subcritical at infinity and …
View article: Energy estimates for seminodal solutions to an elliptic system with mixed couplings
Energy estimates for seminodal solutions to an elliptic system with mixed couplings Open
We study the system of semilinear elliptic equations $$-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
View article: Normalized solutions to a non-variational Schrödinger system
Normalized solutions to a non-variational Schrödinger system Open
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 ca…
View article: Solutions to indefinite weakly coupled cooperative elliptic systems
Solutions to indefinite weakly coupled cooperative elliptic systems Open
We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 + \lambda\beta|u_1|^\alpha|u_2|…
View article: Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems
Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems Open
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 + \s…
View article: Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation
Yamabe systems, optimal partitions, and nodal solutions to the Yamabe equation Open
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 el…