arXiv subjects

Maria Medina

Publications and source records attributed to Maria Medina.

12 recordsLinked to original sources

arXiv2025

Multi-peak solutions for the fractional Schr\"odinger equation with Dirichlet datum

Let $s\in (0,1)$, $\varepsilon>0$ and let $\Omega$ be a bounded smooth domain. Given the problem $$\varepsilon^{2s}(-\Delta)^{s} u + V(x)u = |u|^{p-1}u \quad \mbox{in }\; \Omega,$$ with Dirichlet boundary conditions and $1<p<(n+2s)/(n-2s)$, we analyze the existence of positive multi-peak solutions concentrating, as $\varepsilon\to 0$, to one or several points of $\Omega$. Under suitable conditions on $V$, we construct positive solutions concentrating at any prescribed set of its non degenerate critical points. Furthermore, we prove existence and non existence of clustering phenomena around local maxima and minima of $V$, respectively. The proofs rely on a Lyapunov-Schmidt reduction where three effects need to be controlled: the potential, the boundary and the interaction among peaks. The slow decay of the associated {\it ground-state} demands very precise asymptotic expansions.

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arXiv2022

Segregated solutions for a critical elliptic system with a small interspecies repulsive force

We consider the elliptic system $$-\Delta u_i = u_i^3+\sum\limits_{j=1\atop j\not=i}^{q+1}{ \beta_{ij}}u_i u_j^2\ \hbox{in}\ \mathbb R^4, \ i=1,\dots,q+1.$$ when $\alpha:=\beta_{ij}$ and $\beta:=\beta_{i(q+1)}=\beta_{(q+1)j}$ for any $i,j=1,\dots,q.$ If $\beta<0$ and $|\beta|$ is small enough we build solutions such that each component $u_{1},\dots,u_q$ blows-up at the vertices of $q$ polygons placed in different great circles which are linked to each other, and the last component $u_{q+1}$ looks like the radial positive solution of the single equation.

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arXiv2020

A blow-up phenomenon for a non-local Liouville-type equation

We consider a non-local Liouville equation corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up at a critical point of the harmonic extension of the prescribed curvature function, provided some generic assumptions are satisfied.

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arXiv2020

Large conformal metrics with prescribed Gaussian and geodesic curvatures

We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E. \begin{equation*}\begin{cases}-\Delta u=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\ \partial_\nu u+2=2h(z)e^\frac u2&\hbox{on}\;\partial\mathbb{D}^2,\end{cases} \end{equation*} where $K,h$ are the prescribed curvatures. We construct a family of conformal metrics with curvatures $K_\varepsilon,h_\varepsilon$ converging to $K,h$ respectively as $\varepsilon$ goes to $0$, which blows up at one boundary point under some generic assumptions.

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arXiv2020

Equivalence of weak and viscosity solutions in fractional non-homogeneous problems

We establish the equivalence between the notions of weak and viscosity solutions for non-homogeneous equations whose main operator is the fractional p-Laplacian and the lower order term depends on $x$, $u$ and $D_s^p u$, being the last one a type of fractional derivative.

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arXiv2020

Doubling nodal solutions to the Yamabe equation in $\mathbb{R}^n$ with maximal rank

We construct a new family of entire solutions to the Yamabe equation $$-\Delta u=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u \mbox{ in }\mathcal{D}^{1,2}(\mathbb{R}^n).$$ If $n=3$, our solutions have maximal rank, being the first example in odd dimension. Our construction has analogies with the doubling of the equatorial spheres in the construction of minimal surfaces in $S^3(1)$.

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arXiv2019

Interacting helical vortex filaments in the 3-dimensional Ginzburg-Landau equation

For each given $n\geq 2$, we construct a family of entire solutions $u_\varepsilon (z,t)$, $\varepsilon>0$, with helical symmetry to the 3-dimensional complex-valued Ginzburg-Landau equation \begin{equation*}\nonumber \Delta u+(1-|u|^2)u=0, \quad (z,t) \in \mathbb{R}^2\times \mathbb{R} \simeq \mathbb{R}^3. \end{equation*} These solutions are $2\pi/\varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior as $\varepsilon\to 0$ $$ u_\varepsilon (z,t) \approx \prod_{j=1}^n W\left( z- \varepsilon^{-1} f_j(\varepsilon t) \right), $$ where $W(z) =w(r) e^{i\theta} $, $z= re^{i\theta},$ is the standard degree $+1$ vortex solution of the planar Ginzburg-Landau equation $ \Delta W+(1-|W|^2)W=0 \text{ in } \mathbb{R}^2 $ and $$ f_j(t) = \frac { \sqrt{n-1} e^{it}e^{2 i (j-1)\pi/ n }}{ \sqrt{|\log\varepsilon|}}, \quad j=1,\ldots, n. $$ Existence of these solutions was previously conjectured, being ${\bf f}(t) = (f_1(t),\ldots, f_n(t))$ a rotating equilibrium point for the renormalized energy of vortex filaments there derived, $$ \mathcal W_\varepsilon ( {\bf f} ) :=\pi \int_0^{2\pi} \Big ( \, \frac{|\log \varepsilon|} 2 \sum_{k=1}^n|f'_k(t)|^2-\sum_{j\neq k}\log |f_j(t)-f_k(t)| \, \Big ) \mathrm{d} t, $$ corresponding to that of a planar logarithmic $n$-body problem. These solutions satisfy $$ \lim_{|z| \to +\infty } |u_\varepsilon (z,t)| = 1 \quad \hbox{uniformly in $t$} $$ and have nontrivial dependence on $t$, thus negatively answering the Ginzburg-Landau analogue of the Gibbons conjecture for the Allen-Cahn equation, a question originally formulated by H. Brezis.

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arXiv2017

On viscosity and weak solutions for non-homogeneous p-Laplace equations

In this manuscript we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower order term depending on $x$, $u$ and $\nabla u$. More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending previous results by Juutinen, Lindqvist and Manfredi on the homogeneous case, and Julin and Juutinen for a linear right hand side. Moreover, we provide a converse statement in the full case under extra assumptions on the regularity of the solutions.

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arXiv2017

Principal Eigenvalue of Mixed Problem for the Fractional Laplacian: Moving the Boundary Conditions

We analyze the behavior of the eigenvalues of the following non local mixed problem $\left\{ \begin{array}{rcll} (-\Delta)^{s} u &=& \lambda_1(D) \ u &\inn\Omega,\\ u&=&0&\inn D,\\ \mathcal{N}_{s}u&=&0&\inn N. \end{array}\right $ Our goal is to construct different sequences of problems by modifying the configuration of the sets $D$ and $N$, and to provide sufficient and necessary conditions on the size and the location of these sets in order to obtain sequences of eigenvalues that in the limit recover the eigenvalues of the Dirichlet or Neumann problem. We will see that the non locality plays a crucial role here, since the sets $D$ and $N$ can have infinite measure, a phenomenon that does not appear in the local case (see for example \cite{D,D2,CP}).

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arXiv2016

A fractional elliptic problem in $\mathbb{R}^n$ with critical growth and convex nonlinearities

In this paper we prove the existence of a positive solution of the nonlinear and nonlocal elliptic equation in $\mathbb{R}^n$ \[ (-\Delta)^s u =\varepsilon h u^q+u^{2_s^*-1} \] in the convex case $1\leq q<2_s^*-1$, where $ 2_s^*={2n}/({n-2s}) $ is the critical fractional Sobolev exponent, $(-\Delta)^s$ is the fractional Laplace operator, $\varepsilon$ is a small parameter and $h$ is a given bounded, integrable function. The problem has a variational structure and we prove the existence of a solution by using the classical Mountain-Pass Theorem. We work here with the harmonic extension of the fractional Laplacian, which allows us to deal with a weighted (but possibly degenerate) local operator, rather than with a nonlocal energy. In order to overcome the loss of compactness induced by the critical power we use a Concentration-Compactness principle. Moreover, a finer analysis of the geometry of the energy functional is needed in this convex case.

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arXiv2016

Bifurcation results for a fractional elliptic equation with critical exponent in R^n

In this paper we study some nonlinear elliptic equations in $\R^n$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$ (-Δ)^s u = ε\,h\,u^q + u^p \ {in}\R^n,$$ where $s\in(0,1)$, $n>4s$, $ε>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $0<q<p$ and $h$ is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case $0<q<1$ is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

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arXiv2016

Fractional elliptic problems with critical growth in the whole of $\R^n$

We study the following nonlinear and nonlocal elliptic equation in~$\R^n$ $$ (-Δ)^s u = ε\,h\,u^q + u^p \ {\mbox{ in }}\R^n, $$ where~$s\in(0,1)$, $n>2s$, $ε>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $q\in(0,1)$, and~$h\in L^1(\R^n)\cap L^\infty(\R^n)$. The problem has a variational structure, and this allows us to find a positive solution by looking at critical points of a suitable energy functional. In particular, in this paper, we find a local minimum and a mountain pass solution of this functional. One of the crucial ingredient is a Concentration-Compactness principle. Some difficulties arise from the nonlocal structure of the problem and from the fact that we deal with an equation in the whole of~$\R^n$ (and this causes lack of compactness of some embeddings). We overcome these difficulties by looking at an equivalent extended problem.

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