Existence of bubbling solutions without mass concentration (Q2421929)

Let \(\mathcal{M}=(M,g)\) be a closed Riemann surface with total volume 1. Let \(p>0\) be given. Let \(S=\{q_1,\dots,q_k\}\) be a finite set of vertex points and let \(\delta_{q_i}\) be the associated Dirac measure. Let \(h_*\in C^{2,\sigma}\) be positive and let \(\Delta\) be the Laplacian. Consider the mean field equation \[ \Delta u+p\left(\frac{h_*e^u}{\int h_*e^u}\right)=4\pi\sum_{q_i\in S}\alpha_i(\delta_{q_i}-1)\,. \] Work of \textit{H. Brézis} and \textit{F. Merle} [Commun. Partial Differ. Equations 16, No. 8--9, 1223--1253 (1991; Zbl 0746.35006)] implies any sequence of blow up solutions must exhibit only finitely many points around which their mass concentrates; when the vertex points are non collapsing, the mean field equation possesses bubbling implies mass concentration. \textit{C.-S. Lin} and \textit{G. Tarantello} [C. R., Math., Acad. Sci. Paris 354, No. 5, 493--498 (2016; Zbl 1387.35310)] noted that bubbling implies mass concentration might not hold in general if collapse of singularities occurs. In the present the work, the authors present an example of a non-concentrated bubbling solution of the mean field equation with collapsing singularities. Section 1 contains an introduction to the matter at hand. Section 2 presents preliminaries and introduces estimates illustrating constructing suitable approximate solutions. These approximate solutions are treated in Section 3. A suitable linearized operator is examined in Section 4. Section 5 deals with the invertibility of the linearized operator. Section 6 completes the discussion.