相关论文: Compactness and existence results of the prescribi…

This paper is a continuation of our earlier work "[T. Jin, Y.Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, to appear in J. Eur. Math. Soc.]", where compactness results were…

We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is…

In this work we study an existence and multiplicity result for the following prescribed mean-curvature problem with critical growth $$ \left\{\begin{array}{rl} -\mbox{div}\biggl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\biggl) = \lambda…

Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure.…

Let $(M,g_0)$ be a closed Riemannian manifold of dimension $n \geq 25$ with positive Yamabe invariant $Y(M,g_0)>0$ and positive fourth-order invariant $Y_4(M,g_0)>0$. We show that, arbitrarily $C^1$-close to $g_0$, there exists a Riemannian…

Let $(M, g)$ be a closed Riemannian manifold of dimension $5$. Assume that $(M, g)$ is not conformally equivalent to the round sphere. If the scalar curvature $R_g\geq 0$ and the $Q$-curvature $Q_g\geq 0$ on $M$ with $Q_g(p)>0$ for some…

We prove the compactness of solutions to general fourth order elliptic equations which are L^1-perturbations of the Q-curvature equation on compact Riemannian 4-maniods. Consequently, we prove the global existence and convergence of the…

Let $k\ge1$ be a positive integer and let $P_g$ be the GJMS operator $P_{g}$ of order $2k$ on a closed Riemannian manifold $(M,g)$ of dimension $n>2k$. We investigate the compactness of the set of conformal metrics to $g$ with prescribed…

In this paper we demonstrate that under general conditions there exists a metric in the conformal class of an arbitrary metric on a smooth, closed Riemannian manifold of dimension greater than four such that the $Q$-curvature of the metric…

We prove nonuniqueness results for complete metrics with constant positive fractional curvature conformal to the round metric on $S^n \setminus S^k$, using bifurcation techniques. These are singular (positive) solutions to a non-local…

We consider the problem of prescribing the $Q_{\ gamma}$ curvature on $\mathbb{S}^n$. Using a perturbation method, we obtain existence results for curvatures close to a positive constant.

In this paper, we address the problem of prescribing non-constant $Q$ and boundary $T$ curvatures on the upper hemisphere $\mathbb{S}^4_+\subset \mathbb{R}^5$, via a conformal change of the background metric. This is equivalent to solve a…

Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $\partial M$, respectively, it is natural to ask whether there exist metrics conformal…

Let (M,g) be a compact Riemannian manifold of dimension n \geq 3. The Compactness Conjecture asserts that the set of constant scalar curvature metrics in the conformal class of g is compact unless (M,g) is conformally equivalent to the…

We show a new example of blow-up behaviour for the prescribed $Q$-curvature equation in even dimension $6$ and higher, namely given a sequence $(V_k)\subset C^0(\mathbb{R}^{2n})$ suitably converging we construct {for $n\geq 3$} a sequence…

We consider the problem of prescribing conformally the scalar curvature on compact manifolds of positive Yamabe class in dimension $n \geq 5$. We prove new existence results using Morse theory and some analysis on blowing-up solutions,…

We consider the fractional Nirenberg problem on the standard sphere $\mathbb{S}^n$ with $n\geq 4$. Using the theory of critical points at infinity, we establish an Euler-Hopf type formula and obtain some existence results for curvature…

We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove…

We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $\sigma \in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math.…

We deal with a family of functionals depending on curvatures and we prove for them compactness and semicontinuity properties in the class of closed and bounded sets which satisfy a uniform exterior and interior sphere condition. We apply…