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Salvador Villegas

Publications and source records attributed to Salvador Villegas.

At least 19 recordsLinked to original sources
arXiv2026

Lack of interior $L^q$ bounds for stable solutions to elliptic equations

We consider stable solutions of semilinear elliptic equations of the form $-Δu=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$. In a well-known paper \cite{cfrs}, Cabré, Figalli, Ros-Oton and Serra obtained interior estimates for the $W^{1,2}$-norm of $u$ in terms of the $L^1$-norm of $u$ and proved interior Hölder regularity for dimensions $N\leq 9$. All these results rely on the assumption that $f$ is nonnegative. We show that, for general nonlinearities $f\in C^\infty(\mathbb{R})$, it is impossible, in any dimension $N\geq 1$, to obtain an interior $L^q$ estimate in terms of the $L^p$-norm of $u$ whenever $1\leq p<q\leq \infty$.

math.AP↗
arXiv2026

Regularity of stable radial solutions to semilinear elliptic equations in MEMS problems

This paper investigates the regularity of stable radial solutions to semilinear elliptic equations arising in MEMS problems, modeled by the Dirichlet problem $-Δu=f(u)$ in the unit ball $B_1$, where the nonlinearity $f\in C^1([0,1))$ is nonnegative and satisfies $\int^1_0f(s)\,ds=+\infty$. We focus on the case where $f$ blows up as $u\to 1^{-}$. Micro-electro-mechanical systems (MEMS) are widely used devices in engineering and technology. Our main result establishes for dimensions $2\le n\le 6$, every stable radial solution is regular, meaning $\|u\|_{L^{\infty}(B_1)}<1$. This result gives a positive answer to an open problem posed by Bruera and Cabré concerning the regularity of stable solutions for singular nonlinearities without requiring a Crandall-Rabinowitz type condition, at least in the radial case.

math.AP↗
arXiv2025

A priori estimates of stable solutions of the general Hardy-Henon equation in the ball

This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-Δu=\vert x\vert^αf(u) \mbox{ in } B_1\setminus\lbrace0\rbrace$, where $f\in C^1(\mathbb{R})$ is a general nonlinearity, $α>-2$ and $B_1$ is the unit ball of $\mathbb{R}^N$, $N>1$. We establish the boudness of such solutions for dimensions $2\leq N<10+4α$ and sharp pointwise estimates in the case $N\geq10+4α$. In addition, we provide, for this range of dimensions, a large family of semi-stable radially decreasing unbounded $H^1(B_1)$ solutions.

math.AP↗
arXiv2025

Solutions for autonomous semilinear elliptic equations

We study existence of nontrivial solutions to problem \begin{equation*} \left\lbrace \begin{array}{rcll} -Δu &=& λu+f(u)&\text{ in }Ω,\\ u&=&0&\text{ on }\partial Ω, \end{array}\right. \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $N\geq 1$, $λ\in \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ is any locally Lipschitz function with nonpositive primitive. A complete description is obtained for $N=1$ and partial results for $N\geq 2$.

math.AP↗
arXiv2024

A negative result on regularity estimates on finite radial Morse index solutions to elliptic problems

In the regularity theory of solutions to elliptic partial differential equations often the concept of stability plays the role of a sufficient condition for smoothness. It is a natural question to ask if this holds true for nonstable but finite Morse index solutions. We provide a negative answer showing the existence of sequences of solutions with radial Morse index equal to 1 for which regularity estimates can not be satisfied.

math.AP↗
arXiv2020

A simple proof of the optimal power in Liouville theorems

Consider the equation div$(φ^2 \nabla σ)=0$ in $\mathbb{R}^N,$ where $φ>0$. It is well-known that if there exists $C>0$ such that $\int_{B_R}(φσ)^2 dx\leq CR^2$ for every $R\geq 1$ then $σ$ is necessarily constant. In this paper we prove that this result is not true if we replace $R^2$ by $R^k$ for $k>2$ in any dimension $N$. This question is related to a conjecture by De Giorgi.

math.AP↗
arXiv2020

Behavior near the origin of $f'(u^\ast)$ in radial singular extremal solutions

Consider the semilinear elliptic equation $-Δu=λf(u)$ in the unit ball $B_1\subset \mathbb{R}^N$, with Dirichlet data $u|_{\partial B_1}=0$, where $λ\geq 0$ is a real parameter and $f$ is a $C^1$ positive, nondecreasing and convex function in $[0,\infty)$ such that $f(s)/s\rightarrow\infty$ as $s\rightarrow\infty$. In this paper we study the behavior of $f'(u^\ast)$ near the origin when $u^\ast$, the extremal solution of the previous problem associated to $λ=λ^\ast$, is singular. This answers to an open problems posed by Brezis and Vázquez.

math.AP↗
arXiv2020

Sharp Liouville Theorems

Consider the equation div$(φ^2 \nabla σ)=0$ in $\mathbb{R}^N,$ where $φ>0$. Berestycki, Caffarelli and Nirenberg proved that if there exists $C>0$ such that $\int_{B_R}(φσ)^2 \leq CR^2$ for every $R\geq 1$ then $σ$ is necessarily constant. In this paper we provide necessary and sufficient conditions on $0<Ψ\in C([1,\infty))$ for which this result remains true if we replace $R^2$ with $Ψ(R)$ in any dimension $N$. In the case of the convexity of $Ψ$ for large $R>1$ and $Ψ'>0$, this condition is equivalent to $\displaystyle{\int_1^\infty\frac{1}{Ψ'}=\infty}$.

math.AP↗
arXiv2017

Antisymmetry of solutions for some weighted elliptic problems

This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Nirenberg on the weights is improved to a monotonicity condition. In addition, we provide with a large class of problems where antisymmetry does not hold. Finally, some rather partial extensions in higher dimensions are also given.

math.AP↗
arXiv2014

Sharp estimates of radial minimizers of p-Laplace equations

In this paper we study semi-stable, radially symmetric and decreasing solutions $u\in W^{1,p}(B_1)$ of $-Δ_p u=g(u)$ in $B_1\setminus\{0\}$, where $B_1$ is the unit ball of $\mathbb{R}^N$, $p>1$, $Δ_p$ is the $p-$Laplace operator and $g$ is a general locally Lipschitz function. We establish sharp pointwise estimates for such solutions. As an application of these results, we obtain optimal pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation $-Δ_p u=λf(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, where the nonlinearity $f$ is an increasing $C^1$ function with $f(0)>0$ and $\lim_{t\rightarrow+\infty}{\frac{f(t)}{t^{p-1}}}=+\infty.$ In addition, we provide, for $N\geq p+4p/(p-1)$, a large family of semi-stable radially symmetric and decreasing unbounded $W^{1,p}(B_1)$ solutions.

math.AP↗
arXiv2014

Non-energy semi-stable radial solutions

This paper is devoted to the study of semi-stable radial solutions $u\notin H^1(B_1)$ of $-Δu=f(u) \mbox{in} \overline{B_1}\setminus \{0\}=\{x\in \mathbb{R}^N : 0<\vert x\vert\leq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We establish sharp pointwise estimates for such solutions. In addition, we prove that in dimension $N=2$, any semi-stable radial weak solution of $-Δu=f(u)$, posed in $B_1$ with Dirichlet data $u|_{\partial B_1}=0$, is regular.

math.AP↗
arXiv2014

Dichotomy of stable radial solutions of $-Δu=f(u)$ outside a ball

This paper is devoted to the study of stable radial solutions of $-Δu=f(u) \mbox{ in } \mathbb{R}^N\setminus B_1=\{ x\in \mathbb{R}^N : \vert x\vert\geq 1\}$, where $f\in C^1(\mathbb{R})$ and $N\geq 2$. We prove that such solutions are either large [in the sense that $\vert u(r)\vert \geq M r^{-N/2+\sqrt{N-1}+2}\ $, if $2\leq N\leq 9$; $\vert u(r)\vert \geq M \log (r)\ $, if $N=10$; $\vert u(r)-u_\infty \vert \geq M r^{\-N/2+\sqrt{N-1}+2}\ $, if $N\geq 11$; $\forall r\geq r_0$, for some $M>0$, $r_0\geq 1$] or small [in the sense that $\vert u(r)\vert \leq M\log (r)\ $, if $N=2$; $\vert u(r)-u_\infty \vert \leq M r^{\-N/2-\sqrt{N-1}+2}$;\, if $N\geq 3$; $\forall r\geq 2$, for some $M>0$], where $u_\infty=\lim_{r\rightarrow \infty}u(r)\in [-\infty,+\infty]$. These results can be applied to stable outside a compact set radial solutions of equations of the type $-Δu=g(u) \mbox{ in } \mathbb{R}^N$. We prove also the optimality of these results, by considering solutions of the form $u(r)=r^α$ or $u(r)=\log (r)$, $\forall r\geq 1$, where $α\in \mathbb{R} \setminus \{ 0\}$.

math.AP↗
arXiv2012

Boundedness of the extremal solutions in dimension 4

In this paper we establish the boundedness of the extremal solution u^* in dimension N=4 of the semilinear elliptic equation $-Δu=λf(u)$, in a general smooth bounded domain Omega of R^N, with Dirichlet data $u|_{\partial Ω}=0$, where f is a C^1 positive, nondecreasing and convex function in [0,\infty) such that $f(s)/s\rightarrow\infty$ as $s\rightarrow\infty$. In addition, we prove that, for N>=5, the extremal solution $u^*\in W^{2,\frac{N}{N-2}}$. This gives $u^\ast\in L^\frac{N}{N-4}$, if N>=5 and $u^*\in H_0^1$, if N=6.

math.AP↗
arXiv2011

An applied mathematical excursion through Lyapunov inequalities, classical analysis and differential equations

Several different problems make the study of the so called Lyapunov type inequalities of great interest, both in pure and applied mathematics. Although the original historical motivation was the study of the stability properties of the Hill equation (which applies to many problems in physics and engineering), other questions that arise in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients, etc. lead to the study of $L_p$ Lyapunov inequalities ($1\leq p\leq \infty$) for differential equations. In this work we review some recent results on these kinds of questions which can be formulated as optimal control problems. In the case of Ordinary Differential Equations, we consider periodic and antiperiodic boundary conditions at higher eigenvalues and by using a more accurate version of the Sturm separation theory, an explicit optimal result is obtained. Then, we establish Lyapunov inequalities for systems of equations. To this respect, a key point is the characterization of the best $L^p$ Lyapunov constant for the scalar given problem, as a minimum of some especial (constrained or unconstrained) variational problems defined in appropriate subsets of the usual Sobolev spaces. For Partial Differential Equations on a domain $Ω\subset \real^N$, it is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role in order to obtain nontrivial $L_p$ Lyapunov type inequalities (which are called Sobolev inequalities by many authors). This fact shows a deep difference with respect to the ordinary case. Combining the linear results with Schauder fixed point theorem, we can obtain some new results about the existence and uniqueness of solutions for resonant nonlinear problems for ODE or PDE, both in the scalar case and in the case of systems of equations

math.AP↗
arXiv2011

Lyapunov inequalities for Partial Differential Equations at radial higher eigenvalues

This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \ 1 \leq p \leq +\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\real^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.

math.AP↗
arXiv2010

Stability, Resonance and Lyapunov Inequalities for Periodic Conservative Systems

This paper is devoted to the study of Lyapunov type inequalities for periodic conservative systems. The main results are derived from a previous analysis which relates the best Lyapunov constants to some especial (constrained or unconstrained) minimization problems. We provide some new results on the existence and uniqueness of solutions of nonlinear resonant and periodic systems. Finally, we present some new conditions which guarantee the stable boundedness of linear periodic conservative systems.

math.CA↗
arXiv2009

Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues

This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems, where the Lagrange multiplier Theorem plays a fundamental role. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.

math.AP↗