Related papers: Fourth order equations of critical Sobolev growth.…
In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically $\mathbb{S}^{4}$ or $\mathbb{C}\mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval…
We discuss a large class of conformally invariant curvature energies for immersed hypersurfaces of dimension 4. The class under study includes various examples that have appeared in the recent literature and which arise from different…
On a compact Riemannian manifold, we prove the existence of multiple solutions for an elliptic equation with critical Sobolev growth and critical Hardy potential.
This paper considers the Euler-Lagrange equations satisfied by the critical points of a large class of conformally invariant extrinsic energies for 4-manifolds immersed into Euclidean space (any codimension). Using invariances and Noether's…
We prove some refinements of concentration compactness principle for Sobolev space $W^{1,n}$ on a smooth compact Riemannian manifold of dimension $n$. As an application, we extend Aubin's theorem for functions on $\mathbb{S}^{n}$ with zero…
In this long overdue second installment, we continue to develop the conformal bootstrap program for ${\mathcal N}=4$ superconformal field theories in four dimensions via an analysis of the correlation function of four stress-tensor…
In this paper, we deal with the logarithmic weighted fourth order elliptic equation in the unit disk of $B\subset\R^{4}$: $$\displaystyle(P_{\lambda})~~\Delta(w(x) \Delta u) = \lambda\ f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial…
We provide a full resolution of the Yamabe problem on closed 3-manifolds for Riemannian metrics of Sobolev class $W^{2,q}$ with $q > 3$. This requires developing an elliptic theory for the conformal Laplacian for rough metrics and…
Let $M$ be a compact complex manifold of dimension $n\geq 2$. We prove that for any Hermitian metric $\omega$ on $M$, there exists a unique smooth function $f$ (up to additive constants) such that the conformal metric $\omega_g =e^f \omega$…
In this paper, we establish a weighted Adams' inequality in some appropriate weighted Sobolev space in $\mathbb{R}^4$. Then we give an improvement inequality by proving the concentration-compactness result. In the last part, we consider an…
A conformally invariant generalization of the Willmore energy for compact immersed submanifolds of even dimension in a Riemannian manifold is derived and studied. The energy arises as the coefficient of the log term in the renormalized area…
In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left\{% \begin{array}{ll} (-\Delta)^{s} u-\ds\frac{\mu…
A common problem in physics and engineering is the calculation of the minima of energy functionals. The theory of Sobolev gradients provides an efficient method for seeking the critical points of such a functional. We apply the method to…
In this paper we perform a fine blow-up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere. We derive from this analysis some a…
In this paper we study the topology of conformally compact Einstein 4-manifolds. When the conformal infinity has positive Yamabe invariant and the renormalized volume is also positive we show that the conformally compact Einstein 4-manifold…
This paper deals with some mathematical models arising in the theory of epitaxial growth of crystal. We focalize the study on a stationary problem which presents some analytical difficulties. We study the existence of solutions. The central…
This paper deals with singular/degenerate semilinear critical equations which arise as the Euler-Lagrange equation of Caffarelli-Kohn-Nirenberg inequalities in $\mathbb{R}^d$, with $d\geq 2$. We prove several rigidity results for positive…
We establish a symmetry result for positive entire solutions with a prescribed growth rate to the following fourth order equation on the 3-dimensional hyperbolic space $\mathbb{H}^3$: \[ P_2 u = - u^{-7}, \] where $P_2$ denotes the…
In this paper, without any assumption on $v$ and under the extremely mild assumption $u(x)= O(|x|^{K})$ as $|x|\rightarrow+\infty$ for some $K\gg1$ arbitrarily large, we classify solutions of the following conformally invariant system with…
We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle_{\textbf{g}}u+h.u=f .u^{\frac{n+2}{n-2}} $ on a compact riemannian manifold in the limit case normally not solved by variational…