The Gelfand Problem for the Infinity Laplacian

We study the asymptotic behavior as p → ∞ of the Gelfand problem −Δpu = λeu in Ω ⊂ Rn, u = 0 on ∂Ω. Under an appropriate rescaling on u and λ, we prove uniform convergence of solutions of the Gelfand problem to solutions of min{|∇u|−Λeu, −Δ∞u} = 0 in Ω, u = 0 on ∂Ω. We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of Λ

The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space

© 2021 Elsevier Inc. All rights reserved.We study a natural question that, apparently, has not been well addressed in the literature. Given functions \upsilon with support in the unit ball B1 \subset \mathbb{R}n and with gradient in the Morrey space Mp,\lambda(B1), where 1 \lambda p/(\lambda−p). The function is basically a negative power of the distance to a set of Hausdorff dimension n − \lambda. When \lambda \notin \mathbb{Z}, this set is a fractal. We also make a detailed study of the radially symmetric case, a situation in which the exponent q can go up to np/(\lambda − p).The authors were supported by MICINN grant MTM2017-84214-C2-1-P (Spain). X. Cabré is member of the research group 2017 SGR 1392 (Catalonia). F. Charro was also partially supported by a Juan de la Cierva fellowship and MICINN grants MTM2016-80474-P and PID2019-110712GB-I100 (Spain).Peer ReviewedPostprint (author's final draft

On the existence threshold for positive solutions of p-laplacian equations with a concave-convex nonlinearity

We study the following boundary value problem with a concave-convex nonlinearity: \begin{equation*} \left\{ \begin{array}{r c l l} -\Delta_p u & = & \Lambda\,u^{q-1}+ u^{r-1} & \textrm{in }\Omega, \\ u & = & 0 & \textrm{on }\partial\Omega. \end{array}\right. \end{equation*} Here $\Omega \subset \mathbb{R}^n$ is a bounded domain and $1<q<p<r<p^*$. It is well known that there exists a number $\Lambda_{q,r}>0$ such that the problem admits at least two positive solutions for $0<\Lambda<\Lambda_{q,r}$, at least one positive solution for $\Lambda=\Lambda_{q,r}$, and no positive solution for $\Lambda > \Lambda_{q,r}$. We show that $$\lim_{q \to p} \Lambda_{q,r} = \lambda_1(p),$$ where $\lambda_1(p)$ is the first eigenvalue of the p-laplacian. It is worth noticing that $\lambda_1(p)$ is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case $q=p$

Asymptotic Mean-Value Formulas for Solutions of General Second-Order Elliptic Equations

We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and k-Hessian operators