Existence and local uniqueness of bubbling solutions for poly-harmonic equations with critical growth

Abstract

\begin{abstract} We consider the following poly-harmonic equations with critical exponents: \begin{equation}\label{P} (-Δ)^m u =K(y)u^{\frac{N+2m}{N-2m}},\;\;\; u>0\;\;\;\hbox{in} \mathbb{R}^N, \end{equation} where $N>2m+2,m\in {\mathbb{N}}_{+},K(y)$ is positive and periodic in its first $k$ variables $(y_1,\cdots ,y_k)$, $1\le k<\frac{N-2m}{2}$. Under some conditions on $K(y)$ near its critical point, we prove not only that problem~\eqref{P} admits solutions with infinitely many bubbles, but also that the bubbling solutions obtained in our existence result are locally unique. This local uniqueness result implies that some bubbling solutions preserve the symmetry of the scalar curvature $K(y).$

39 pages

References in corpus (1)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)

Cited by in corpus (2)

• Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities
• Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations