Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations

We investigate the asymptotic behavior of solutions of anisotropic equations of the form $-\sum_{i=1}^n\partial_{x_i}(\left|\partial_{x_i}u\right|^{p_i-2}\partial_{x_i}u)=f(x,u)$ in $\mathbb{R}^n$, where $p_i>1$ for all $i=1,\dotsc,n$ and $f$ is a Caratheodory function with critical Sobolev growth. This problem arises in particular from the study of extremal functions for a class of anisotropic Sobolev inequalities. We establish decay estimates for the solutions and their derivatives, and we bring to light a vanishing phenomenon which occurs when the maximum value of the exponents $p_i$ exceeds a critical value.Comment: Final version to appear in Advances in Mathematic

Sign-changing blow-up for scalar curvature type equations

Given $(M,g)$ a compact Riemannian manifold of dimension $n\geq 3$, we are interested in the existence of blowing-up sign-changing families (\ue)_{\eps>0}\in C^{2,\theta}(M), $\theta\in (0,1)$, of solutions to \Delta_g \ue+h\ue=|\ue|^{\frac{4}{n-2}-\eps}\ue\hbox{ in }M\,, where $\Delta_g:=-\hbox{div}_g(\nabla)$ and $h\in C^{0,\theta}(M)$ is a potential. We prove that such families exist in two main cases: in small dimension $n\in \{3,4,5,6\}$ for any potential $h$ or in dimension $3\leq n\leq 9$ when h\equiv\frac{n-2}{4(n-1)}\Scal_g. These examples yield a complete panorama of the compactness/noncompactness of critical elliptic equations of scalar curvature type on compact manifolds. The changing of the sign is necessary due to the compactness results of Druet and Khuri--Marques--Schoen

Sign-changing solutions to elliptic second order equations: glueing a peak to a degenerate critical manifold

We construct blowing-up sign-changing solutions to some nonlinear critical equations by glueing a standard bubble to a degenerate function. We develop a method based on analyticity to perform the glueing when the critical manifold of solutions is degenerate and no Bianchi--Egnell type condition holds.Comment: Final version to appear in "Calculus of Variations and PDEs

Examples of non-isolated blow-up for perturbations of the scalar curvature equation on non locally conformally flat manifolds

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.Comment: Final version to appear in J. of Differential Geometry. References updated, details adde

A note on the classification of positive solutions to the critical p-Laplace equation in Rn\mathbb{R}^n

In this note, we obtain a classification result for positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$ with $n\ge4$ and $p>p_n$ for some number $p_n\in\left(\frac{n}{3},\frac{n+1}{3}\right)$ such that $p_n\sim\frac{n}{3}+\frac{1}{n}$, which slightly improves upon a similar result recently obtained by Ou under the condition $p\ge\frac{n+1}{3}$

Existence and regularity for critical anisotropic equations with critical directions

International audienceWe establish existence and regularity results for doubly critical anisotropic equations in domains of the Euclidean space. In particular, we answer a question posed by Fragala-Gazzola-Kawohl [24] when the maximum of the anisotropic con guration coincides with the critical Sobolev exponent

Liouville-type results for the CR Yamabe equation in the Heisenberg group

We obtain Liouville-type results for solutions to the CR Yamabe equation in $\mathbb{H}^n$, which extend a result obtained by Jerison and Lee for solutions in $L^{2+2/n}(\mathbb{H}^n)$. We obtain our results under either pointwise conditions or integral conditions at infinity. In particular, our results hold for all bounded solutions when $n=2$ and solutions satisfying a pointwise decay assumption when $n\ge3$. The proofs rely on integral estimates combined with a suitable divergence formula.Comment: Theorem 1.1 is improved and Theorem 1.4 is adde