arXiv subjects

Carolina Rey

Publications and source records attributed to Carolina Rey.

5 recordsLinked to original sources

arXiv2025

Concentration on the Boundary and Sign-Changing Solutions for a Slightly Subcritical Biharmonic Problem

We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} Δ(a(x)Δu) = a(x) \left\vert u \right\vert^{p-2-ε} u \ \text{ in } \ Ω, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial Ω, \hspace{0.6cm} Δu = 0 \ \text{ on } \ \partial Ω, \end{array}\end{equation*} where $Ω$ is a smooth, bounded domain in $\mathbb{R}^N$ with $N \geq 5$. Here, $p := \frac{2N}{N-4}$ is the Sobolev critical exponent for the embedding $H^2 \cap H_0^1(Ω) \hookrightarrow L^p(Ω)$, and $a \in C^2(\overlineΩ)$ is a strictly positive function on $\overlineΩ$. We establish sufficient conditions on the function $a$ and the domain $Ω$ for this problem to admit both positive and sign-changing solutions with an explicit asymptotic profile. These solutions concentrate and blow up at a point on the boundary $\partial Ω$ as $ε\to 0$. The proofs of the main results rely on the Lyapunov-Schmidt finite-dimensional reduction method.

math.AP↗

arXiv2025

Einstein multiply warped products and generalized Kasner manifolds with multidimensional base

The purpose of this paper is to provide conditions for the existence or non existence of non trivial Einstein multiply warped products, specially of generalised Kasner type; as well as to show estimates of the Einstein parameter that condition the existence of such metrics.

math.DG↗

arXiv2024

Concentration Phenomena for Conformal Metrics with Constant $Q$-Curvature

Let $(M,g)$ be an analytic Riemannian manifold of dimension $n \geq 5$. In this paper, we consider the so-called constant $Q$-curvature equation \[ \varepsilon^4Δ_{g}^2 u -\varepsilon^2 b Δ_{g} u +a u = u^{p} , \qquad \text{in } M, \quad u>0, \quad u\in H^2_g(M) \] where $a,b$ are positive constants such that $b^2-4 a>0$, $p$ is a sub-critical exponent $1 0$ is small enough, then positive solutions to the above constant $Q$-curvature equation are generated by a maximum or minimum point of the function $τ_g$, given by \[ τ_g(ξ):= \sum_{i, j=1}^{n} \frac{\partial^{2} g_ξ^{i i}}{\partial z_{j}^{2}}(0), \] where $g_ξ^{i j}$ denotes the components of the inverse of the metric $g$ in geodesic normal coordinates. This result shows that the geometry of $M$ plays a crucial role in finding solutions to the equation above and provides a metric of constant $Q$-curvature on a product manifold of the form $(M\times X, g+\varepsilon^2 h)$ where $(M,g)$ is flat and closed, and $(X,h)$ any $m$-dimensional Einstein Riemannian manifold, $m\geq 3$.

math.DG↗

arXiv2023

Multiplicity results for constant Q-curvature conformal metrics

We prove that certain subcritical Paneitz-Branson type equations on a closed Riemannian manifold $(M,g)$ have at least $\mathrm{Cat}(M)$ positIve solutions, where $\mathrm{Cat}(M)$ is the Lusternik-Schnirelmann category of $M$. This implies that if $(X,h)$ is a closed positive Einstein manifold then for $\ep >0$ small enough there are at least $\mathrm{Cat}(M)$ metrics of constant $Q$-curvature in the conformal class of the Riemannian product $g+\ep h$.

math.DG↗

arXiv2022

Non-local equations and optimal Sobolev inequalities on compact manifolds

This paper deals with fractional Sobolev spaces on a compact Riemannian manifold. We prove a Sobolev inequality in the critical range with an optimal constant for these fractional Sobolev spaces. We use this result to study the existence of a non-trivial solution for equations driven by a non-local integro-differential operator $\mathcal{L}_{\mathcal{K}}$ with critical non-linearity.

math.AP↗