YanYan Li

Remark on some conformally invariant integral equations: the method of moving spheres

where a > 0 is some constant and x ∈ R. Hypothesis (2) was removed by Caffarelli, Gidas and Spruck in [8]; this is important for applications. Such Liouville type theorems have been extended to general conformally invariant fully nonlinear equations by Li and Li ([24]–[27]); see also related works of Viaclovsky ([40]–[41]) and Chang, Gursky and Yang ([13]–[14]). The method used in [21], as well as in much of the above cited work, is the method of moving planes. The method of moving planes has become a very powerful tool in the study of nonlinear elliptic equations; see Aleksandrov [1], Serrin [38], Gidas, Ni and Nirenberg [21]–[22], Berestycki and Nirenberg [2], and others. In [30], Li and Zhu gave a proof of the above mentioned theorem of Caffarelli, Gidas and Spruck using the method of moving spheres (i.e. the method of moving planes together with the conformal invariance), which fully exploits the conformal invariance of the problem and, as a result, captures the solutions directly rather than going through the usual procedure of proving radial symmetry of solutions and then classifying radial solutions. Significant simplifications to the proof in [30] have been made in Li and Zhang [29]. The method of moving spheres has been used in [24]–[27]. Liouville type theorems for various conformally invariant equations have received much attention; see, in addition to the above cited papers, [23], [17], [15], [33], [42] and [43].

On a min-max procedure for the existence of a positive solution for certain scalar field equations in RN.

In this paper we mainly introduce a min-max procedure to prove the existence of positive solutions for certain semilinear elliptic equations in RN.

On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions

We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem.

YAMABE TYPE EQUATIONS ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS

A theorem of Escobar and Schoen asserts that on a positive three dimensional smooth compact Riemannian manifold which is not conformally equivalent to the standard three dimensional sphere, a necessary and sufficient condition for a C2 function K to be the scalar curvature function of some conformal metric is that K is positive somewhere. We show that for any positive C2 function K, all such metrics stay in a compact set with respect to C3 norms and the total Leray-Schauder degree of all solutions is equal to -1. Such existence and compactness results no longer hold in such generality in higher dimensions or on manifolds conformally equivalent to standard three dimensional spheres. The results are also established for more general Yamabe type equations on three dimensional manifolds.

Morse and Melnikov functions for NLS Pde's

AbstractThe theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat linear operator

The Yamabe problem on manifolds with boundary: Existence and compactness results

\hat L

Existence of solutions for semilinear elliptic equations with indefinite linear part

, is developed in sufficient detail for later use in studies of perturbations of the NLS equation. “Counting lemmas” for the non-selfadjoint operator