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Grey Ercole

Publications and source records attributed to Grey Ercole.

At least 19 recordsLinked to original sources
arXiv2024

Asymptotics for Sobolev extremals: the hyperdiffusive case

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $p>N$ and $1\leq q(p)<\infty$ set \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(\Omega)\text{ \ and \ }\int_{\Omega }\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} \] and let $u_{p,q(p)}$ denote a corresponding positive extremal function. We show that if $\lim\limits_{p\rightarrow\infty}q(p)=\infty$, then $\lim\limits_{p\rightarrow\infty}\lambda_{p,q(p)}^{1/p}=\left\Vert d_{\Omega }\right\Vert _{\infty}^{-1}$, where $d_{\Omega}$ denotes the distance function to the boundary of $\Omega.$ Moreover, in the hyperdiffusive case: $\lim\limits_{p\rightarrow\infty}\frac{q(p)}{p}=\infty,$ we prove that each sequence $u_{p_{n},q(p_{n})},$ with $p_{n}\rightarrow\infty,$ admits a subsequence converging uniformly in $\overline{\Omega}$ to a viscosity solution to the problem \[ \left\{ \begin{array} [c]{lll} -\Delta_{\infty}u=0 & \text{in} & \Omega\setminus M\\ u=0 & \text{on} & \partial\Omega\\ u=1 & \text{in} & M, \end{array} \right. \] where $M$ is a closed subset of the set of all maximum points of $d_{\Omega}.$

math.AP↗
arXiv2023

The Cheeger constant as limit of Sobolev-type constants

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $1<p<N$ and $0<q(p)<p^{\ast}:=\frac{Np}{N-p}$ let \[ \lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(\Omega)\text{ \ and \ }\int_{\Omega }\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} . \] We prove that if $\lim_{p\rightarrow1^{+}}q(p)=1,$ then $\lim_{p\rightarrow 1^{+}}\lambda_{p,q(p)}=h(\Omega)$, where $h(\Omega)$ denotes the Cheeger constant of $\Omega.$ Moreover, we study the behavior of the positive solutions $w_{p,q(p)}$ to the Lane-Emden equation $-\operatorname{div}(\left\vert \nabla w\right\vert ^{p-2}\nabla w)=\left\vert w\right\vert ^{q-2}w,$ as $p\rightarrow1^{+}.$

math.AP↗
arXiv2023

The limiting behavior of solutions to p-Laplacian problems with convection and exponential terms

We consider, for $a,l\geq1,$ $b,s,\alpha>0,$ and $p>q\geq1,$ the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded domain $\Omega\subset\mathbb{R}^{N}.$ We prove that under certain setting of the parameters $\lambda,$ $\beta$ and $m$ the problem admits at least one positive solution. Using this result we prove that if $\lambda,\beta>0$ are arbitrarily fixed and $m$ is sufficiently small, then the problem has a positive solution $u_{p},$ for all $p$ sufficiently large. In addition, we show that $u_{p}$ converges uniformly to the distance function to the boundary of $\Omega,$ as $p\rightarrow\infty.$ This convergence result is new for nonlinearities involving a convection term.

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arXiv2023

On a global gradient estimate in $p$-Laplacian problems

We make explicit the $p$-dependence of $C$ in the gradient estimate $\left\Vert \nabla u\right\Vert _{\infty}^{p-1}\leq C\left\Vert f\right\Vert _{N,1}$ by Cianchi and Maz'ya (2011). In such inequality, the constant $C$ is uniform with respect to $f\in L^{N,1}(\Omega),$ and $u$ is the weak solution to the Poisson equation $-\operatorname{div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=f$ in a bounded domain $\Omega\subset\mathbb{R}^{N},$ $N\geq3,$ coupled with either Neumann or Dirichlet homogeneous boundary conditions. The case $N=2$ with $f\in L^{q}(\Omega),$ for some $q>2,$ is also considered .

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arXiv2020

An optimal pointwise Morrey-Sobolev inequality

Let $Ω$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq1.$ For each $p>N$ we study the optimal function $s=s_{p}$ in the pointwise inequality \[ \left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert _{L^{p}(Ω)},\quad\forall\,(x,v)\in\overlineΩ\times W_{0}% ^{1,p}(Ω). \] We show that $s_{p}\in C_{0}^{0,1-(N/p)}(\overlineΩ)$ and that $s_{p}$ converges pointwise to the distance function to the boundary, as $p\rightarrow\infty.$ Moreover, we prove that if $Ω$ is convex, then $s_{p}$ is concave and has a unique maximum point.

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arXiv2020

Torsion functions and the Cheeger problem: a fractional approach

Let $Ω$ be a Lipschitz bounded domain of $\mathbb{R}^N $, $N\geq2$. The fractional Cheeger constant $h_s (Ω)$, $0<s<1$, is defined by \[h_s(Ω)=\inf_{E\subsetΩ}\frac{P_s(E)}{|E|},\: \text{ where } \: P_s (E)=\int_{\mathbb{R}^N }\int_{\mathbb{R}^N }\frac{|χ_{E}(x)-χ_{E}(y)|}{|x-y|^{N+s}} dx dy,\] with $χ_{E}$ denoting the characteristic function of the smooth subdomain $E$. The main purpose of this paper is to show that \[\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^{\infty}(Ω)}^{1-p}=h_s (Ω)=\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^1(Ω)}^{1-p},\] where $ϕ_p^s$ is the fractional $(s,p)$-torsion function of $Ω$, that is, the solution of the Dirichlet problem for the fractional $p$-Laplacian: $-(Δ)_p^s\,u=1$ in $Ω$, $u=0$ in $\mathbb{R}^N \setminusΩ$. For this, we derive suitable bounds for the first eigenvalue $λ_{1,p}^s(Ω)$ of the fractional $p$-Laplacian operator in terms of $ϕ_p^s$. We also show that $ϕ_p^s$ minimizes the $(s,p)$-Gagliardo seminorm in $\mathbb{R}^N $, among the functions normalized by the $L^1$-norm.

math.AP↗
arXiv2019

On the behavior of least energy solutions of a fractional $(p,q(p))$-Laplacian problem as p goes to infinity

We study the behavior as $p\rightarrow\infty$ of $u_{p},$ a positive least energy solution of the problem \[ \left\{\begin{array} [c]{lll} \left[ \left( -Δ_{p}\right) ^α+\left( -Δ_{q(p)}\right) ^β\right] u=μ_{p}\left\Vert u\right\Vert _{\infty}^{p-2} u(x_{u})δ_{x_{u}} & \mathrm{in} & Ω\\ u=0 & \mathrm{in} & \mathbb{R}^{N}\setminusΩ\\ \left\vert u(x_{u})\right\vert =\left\Vert u\right\Vert _{\infty}, & & \end{array} \right. \] where $Ω\subset\mathbb{R}^{N}$ is a bounded, smooth domain, $δ_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & 0<β<α<1\\ (1,\infty) & \mathrm{if} & 0<α<β<1 \end{array} \right. \] and \[ \lim_{p\rightarrow\infty}\sqrt[p]{μ_{p}}>R^{-α}, \] with $R$ denoting the inradius of $Ω.$

math.AP↗
arXiv2019

Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems

Let $Ω$ be a smooth, bounded domain of $\mathbb{R}^{N}$, $ω$ be a positive, $L^{1}$-normalized function, and $0<s<1<p.$ We study the asymptotic behavior, as $p\rightarrow\infty,$ of the pair $\left( \sqrt[p]{Λ_{p}% },u_{p}\right) ,$ where $Λ_{p}$ is the best constant $C$ in the Sobolev type inequality \[ C\exp\left( \int_Ω(\log\left\vert u\right\vert ^{p})ω\mathrm{d}x\right) \leq\left[ u\right] _{s,p}^{p}\quad\forall\,u\in W_{0}^{s,p}(Ω) \] and $u_{p}$ is the positive, suitably normalized extremal function corresponding to $Λ_{p}$. We show that the limit pairs are closely related to the problem of minimizing the quotient $\left\vert u\right\vert _{s}/\exp\left( \int_Ω(\log\left\vert u\right\vert )ω\mathrm{d}x\right) ,$ where $\left\vert u\right\vert _{s}$ denotes the $s$-Hölder seminorm of a function $u\in C_{0}^{0,s}(\overlineΩ).$

math.AP↗
arXiv2019

Asymptotic behavior as $p\rightarrow\infty$ of least energy solutions of a $(p,q(p))$-Laplacian problem

\[ \left\{ \begin{array} [c]{lll} -\left( Δ_{p}+Δ_{q(p)}\right) u=λ_{p}\left\vert u(x_{u})\right\vert ^{p-2}u(x_{u})δ_{x_{u}} & \mathrm{in} & Ω\\ u=0 & \mathrm{on} & \partialΩ, \end{array} \right. \] where $x_{u}$ is the (unique) maximum point of $\left\vert u\right\vert ,$ $δ_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \[ \lim_{p\rightarrow\infty}\frac{q(p)}{p}=Q\in\left\{ \begin{array} [c]{lll} (0,1) & \mathrm{if} & N 0$ is such that \[ \min\left\{ \frac{\left\Vert \nabla u\right\Vert _{\infty}}{\left\Vert u\right\Vert _{\infty}}:0\not \equiv u\in W^{1,\infty}(Ω)\cap C_{0}(\overlineΩ)\right\} \leq\lim_{p\rightarrow\infty}(λ_{p})^{\frac{1}{p}}<\infty. \]

math.AP↗
arXiv2017

An inverse iteration method for obtaining q-eigenpairs of the p-Laplacian in a general bounded domain

Let $Ω$ be a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2$, and consider the eigenvalue problem: $-Δ_{p}u=λ\left| u\right| _{L^{q}(Ω)}^{p-q}\left| u\right| ^{q-2}u$ in $Ω,$ $u=0$ on $\partialΩ,$ where $p>1$, $1\leq q<p^{\star}$ and $p^{\star}$ is the critical exponent of the Sobolev embedding $W_{0}^{1,p}(Ω)\hookrightarrow L^{q}(Ω)$. Two sequences, $\left( λ_{n}\right)_{n\in N}% \subset(0,\infty)$ and $\left(w_{n}\right) _{n\in N}\subset W_{0}^{1,p}(Ω)$, are built by means of an inverse iteration scheme starting from an arbitrary function $u_{0}\in L^{q}(Ω)\backslash\left\{ 0\right\} $. It is shown that $\left( λ_{n}\right) _{n\in N}$ converges monotonically to an eigenvalue $λ\geqλ_{q}$, with $λ_{q}$ denoting the first eigenvalue. It is also proved that there exists a subsequence $\left( w_{n_{j}}\right) _{j\in\mathbb{N}}$ converging in $W_{0}^{1,p}(Ω)$ to an eigenfunction $w$ corresponding to $λ$. The advantage of this method is that it can be used to find eigenvalues other than $λ_{q}$.

math.AP↗
arXiv2017

Solving an abstract nonlinear eigenvalue problem by the inverse iteration method

Let $\left( X,\left\Vert \cdot\right\Vert_{X}\right) $ and $\left( Y,\left\Vert \cdot\right\Vert_{Y}\right) $ be Banach spaces over $\mathbb{R},$ with $X$ uniformly convex and compactly embedded into $Y.$ The inverse iteration method is applied to solve the abstract eigenvalue problem $A(w)=λ\left\Vert w\right\Vert_{Y}^{p-q}B(w),$ where the maps $A:X\rightarrow X^{\star}$ and $B:Y\rightarrow Y^{\star}$ are homogeneous of degrees $p-1$ and $q-1,$ respectively.

math.AP↗
arXiv2017

Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

In this work, we study the of positive ground state solution for the semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -Δu=u^{p(x)-1},\quad u>0 & \mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\ u\in D_{0}^{1,2}(G), & \end{array} \right. $$ where $G$ is either $\mathbb{R}^{N}$ or a bounded domain, and $p:G\rightarrow \mathbb{R}$ is a continuous function assuming critical and subcritical values.

math.AP↗
arXiv2017

Minimization of quotients with variable exponents

Let $Ω$ be a bounded domain of $\mathbb{R}^{N}$, $p\in C^{1}(\overlineΩ),$ $q\in C(\overlineΩ)$ and $l,j\in\mathbb{N}.$ We describe the asymptotic behavior of the minimizers of the Rayleigh quotient $\frac{\Vert\nabla u\Vert_{lp(x)}}{\Vert u\Vert_{jq(x)}}$, first when $j\rightarrow\infty$ and after when $l\rightarrow\infty.$

math.AP↗
arXiv2016

On a singular minimizing problem

We study a minimizing problem associated with the singular problem \[ \left\{ \begin{array} [c]{ll} -\operatorname{div}\left( \left\vert \nabla u\right\vert ^{p-2}\nabla u\right) =λu^{-1} & \mathrm{in\ }Ω\\ u>0 & \mathrm{in\ }Ω\\ u=0 & \mathrm{on\ }\partialΩ, \end{array} \right. \] where $p>1$, $λ>0$ and $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2.$ A new log-Sobolev type inequality is proved and the corresponding best constant is identifyied.

math.AP↗
arXiv2015

Asymptotics for the best Sobolev constants and their extremal functions

Let $Ω$ be a bounded domain of $\mathbf{R}^{N},$ $N\geq2.$ Let, for $p>N,$ \[ Λ_{p}(Ω):=\inf\left\{ \left\Vert \nabla u\right\Vert _{p}^{p}:u\in W_{0}^{1,p}(Ω)\quad and\quad\left\Vert u\right\Vert _{\infty}=1\right\} . \] We first prove that \[ \lim_{p\rightarrow\infty}Λ_{p}(Ω)^{\frac{1}{p}}=\frac{1}{\left\Vert ρ\right\Vert _{\infty}}, \] where $ρ$ denotes the distance function to the boundary. Then, we show that, up to subsequences, the extremal functions of $Λ_{p}(Ω)$ converge (as $p\rightarrow\infty$) to the viscosity solutions of a specific Dirichlet problem involving the infinity Laplacian in the punctured $Ω.$

math.AP↗
arXiv2012

A quasilinear problem with fast growing gradient

In this paper we consider the following Dirichlet problem for the $p$-Laplacian in the positive parameters $λ$ and $β$: [{{array} [c]{rcll}% -Δ_{p}u & = & λh(x,u)+βf(x,u,\nabla u) & \text{in}Ωu & = & 0 & \text{on}\partialΩ, {array}. \hfill] where $h,f$ are continuous nonlinearities satisfying $0\leqω_{1}(x)u^{q-1}\leq h(x,u)\leqω_{2}(x)u^{q-1}$ with $1 0$, and $Ω$ is a bounded domain of $\mathbb{R}^{N},$ $N\geq3.$ The functions $ω_{i}$, $1\leq i\leq3$, are nonnegative, continuous weights in $\barΩ$. We prove that there exists a region $\mathcal{D}$ in the $λβ$-plane where the Dirichlet problem has at least one positive solution. The novelty in this paper is that our result is valid for nonlinearities with growth higher than $p$ in the gradient variable.

math.AP↗
arXiv2012

On the resonant Lane-Emden problem for the p-Laplacian

We study the positive solutions of the Lane-Emden equation $-Δ_{p}u=λ_{p}|u|^{q-2}u$ in $Ω$ with homogeneous Dirichlet boundary conditions, where $Ω\subset\mathbb{R}^{N}$ is a bounded and smooth domain, $N\geq2,$ $λ_{p}$ is the first eigenvalue of the $p$-Laplacian operator $Δ_{p}$ and $q$ is close to $p>1.$ We prove that any family of positive solutions of this problem converges in $C^{1}(\barΩ)$ to the function $θ_{p}e_{p}$ when $q\rightarrow p,$ where $e_{p}$ is the positive and $L^{\infty}$-normalized first eigenfunction of the $p$-Laplacian and $θ_{p}:=\exp(|e_{p}|_{L^{p}(Ω)}^{-p}\int_Ωe_{p}% ^{p}|\ln e_{p}|dx).$ A consequence of this result is that the best constant of the immersion $W_{0}^{1,p}(Ω)\hookrightarrow L^{q}(Ω)$ is differentiable at $q=p.$ Previous results on the asymptotic behavior (as $q\rightarrow p$) of the positive solutions of the non-resonant Lane-Emden problem (i.e. with $λ_{p}$ replaced by a positive $λ\neqλ_{p}$) are also generalized to the space $C^{1}% (\barΩ)$ and to arbitrary families of these solutions. Moreover, if $u_{λ,q}$ denotes a solution of the non-resonant problem for an arbitrarily fixed $λ>0,$ we show how to obtain the first eigenpair of the $p$-Laplacian as the limit in $C^{1}(\barΩ),$ when $q\rightarrow p$, of a suitable scaling of the pair $(λ,u_{λ,q}).$ For computational purposes the advantage of this approach is that $λ$ does not need to be close to $λ_{p}.$ Finally, an explicit estimate involving $L^{\infty}$ and $L^{1}$ norms of $u_{λ,q}$ is also deduced using set level techniques.

math.AP↗