Related papers: Decay of extremals of Morrey's inequality

We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ which includes $\log |x|$ and $|x|^\alpha$, where $-1 < \alpha < 1$. We also give periodic versions of these results in which the majorants…

In this paper we provide the classification of positive solutions to the critical $p-$Laplace equation on $\mathbb{R}^n$, for $1<p<n$, possibly having infinite energy. If $n=2$, or if $n=3$ and $\frac 32<p<2$ we prove rigidity without any…

Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption $$\partial_t u - \mathrm{div}(|\nabla u|^{p-2}\nabla u) +|\nabla u|^{q}=0 \qquad {\rm in} \…

We study the Lane-Emden problem \[\begin{cases} -\Delta u_p =|u_p|^{p-1}u_p&\text{in}\quad \Omega, u_p=0 &\text{on}\quad\partial\Omega, \end{cases}\] where $\Omega\subset\mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently…

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{on} \;\; \mathbb{R}^n \] in the…

In this short note we consider an unconventional overdetermined problem for the torsion function: let $n\geq 2$ and $\Omega$ be a bounded open set in $\mathbb{R}^n$ whose torsion function $u$ (i.e. the solution to $\Delta u=-1$ in $\Omega$,…

We study the behavior of the smallest possible constants $d(a,b)$ and $d_n$ in Hardy inequalities $$ \int_a^b\left(\frac{1}{x}\int_a^xf(t)dt\right)^p\,dx\leq d(a,b)\,\int_a^b [f(x)]^p dx $$ and $$…

In 2003, Del Pino and Dolbeault [14] and Gentil [19] investigated, independently, best constants and extremals associated to Euclidean Lp-entropy inequalities for p > 1. In this work, we present some contributions in the Riemannian context.…

In this paper we prove some new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities, in any dimension larger or equal than two.

This article investigates the Fourier extension operator associated with the fractional surface $(\xi,|\xi|^{\alpha})$ for $\alpha\geq 2$. We show that the relevant $L^p\to L^q$ Fourier extension inequality possesses extremals for all…

We study the large time behavior of solutions of the PDE $|v_t|^{p-2}v_t=\Delta_p v$. A special property of this equation is that the Rayleigh quotient $\int_{\Omega}|Dv(x,t)|^pdx /\int_{\Omega}|v(x,t)|^pdx$ is nonincreasing in time along…

We study the generalized existence of extremizers for the sharp $p$-Sobolev inequality on noncompact Riemannian manifolds in connection with nonnegative curvature and Euclidean volume growth assumptions. Assuming a nonnegative Ricci…

We study strictly positive solutions to the critical Laplace equation \[ - \Delta u = n(n-2) u^{\frac{n+2}{n-2}}, \] decaying at most like $d(o, x)^{-(n-2)/2}$, on complete noncompact manifolds $(M, g)$ with nonnegative Ricci curvature, of…

In this article we prove the global existence of weak solutions to an initial boundary value problem with an exponential and p-Laplacian nonlinearity. The equation is a continuum limit of a family of kinetic Monte Carlo models of crystal…

The classical Alexandrov estimate controls the oscillation of a convex function by the mass of its associated Monge-Amp\`ere measure and yields, for two convex functions of $n$ variables with the same boundary values, a sup-norm bound with…

We develop a thermodynamic formalism for a strongly dissipative H\'enon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$…

A necessary and sufficient condition on the precompactness of extremal sequences for one dimensional $\alpha$-Strichartz inequalities, equivalently $\alpha$-Fourier extension estimates, is established based on the profile decomposition…

We prove that the Dirichlet problem for degenerate elliptic equations $\mathrm{div}(A \nabla u) = 0$ in the upper half-space $(x,t)\in \mathbb{R}^{n+1}_+$ is solvable when $n\geq2$ and the boundary data is in $L^p_\mu(\mathbb{R}^n)$ for…

In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the…

\small{In 2004, Del Pino and Dolbeault \cite{DPDo} and Gentil \cite{G} investigated, independently, best constants and extremals associated to sharp Euclidean $L^p$-entropy inequalities. In this work, we present some important advances in…