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Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets. (English) Zbl 0879.58019
Given a 2-dimensional Riemannian manifold $$M$$ with (possibly empty) boundary, a closed Riemannian manifold $$N\subset \mathbb{R}^k$$, and $$g: \partial M \to N$$ a Lipschitz mapping, let $$H_g' (M,N)$$ be the set of all $$u\in H' (M,\mathbb{R}^k)$$ with $$u(x) \in N$$ for a.e. $$x\in M$$ and $$u|_{\partial M} =g$$. A Palais-Smale sequence for the Dirichlet energy $$E$$ on $$H_g'(M,N)$$ is a sequence $$\{u_n\} \subset H_g' (M,N)$$ such that $$\limsup_{n \to \infty} E(u_n)$$ is finite and $$E'(u_n) \to 0$$ in $$H^{-1} (M,TN)$$ as $$n\to\infty$$, where $$E'$$ is the derivative of $$E$$ and $$H^- (M, \cdot)$$ is the dual of $$H_0' (M, \cdot)$$.
The author’s main result is the following: Given a Palais-Smale sequence $$\{u_n\}$$ satisfying $$-\Delta u_n= A(u_n) (Du_n, Du_n)+ h_n$$, where $$A$$ is the second fundamental form of $$N$$ in $$\mathbb{R}^k$$, $$h_n \to 0$$ in $$H^{-1} (M,\mathbb{R}^k)$$, and $$\sup_n |h_n |_{L^2} <\infty$$, there exist a harmonic map $$u\in C^\infty (M,N) \cap C^0 (\overline M,N)$$ and a finite number $$\ell$$ of bubbles $$\omega_i \in C^\infty (S^2,N)$$, points $$\{a_n^i\} \subset M$$, and scalars $$\{\lambda^i_n\}$$ with $$1\leq i\leq \ell$$ such that we can extract a subsequence $$\{u_n\}$$ such that
$\lim_{n\to \infty} E(u_n)= E(u)+ \sum^\ell_{i=1} E(\omega_i); \tag{1}$
$\text{For } i\neq j, \max \left\{{\lambda^i_n \over\lambda^j_n}, {\lambda^j_n \over \lambda^i_n}, {\text{dis} (a^i_n, a_n^j) \over \lambda^i_n + \lambda^j_n} \right\} \to\infty \quad \text{as } n\to\infty; \tag{2}$
$\left|u_n- u-\sum^\ell_{i =1} \left[\omega_i \left({-a^i_n \over \lambda^i_n} \right)- \omega _i (\infty) \right] \right |_{H'} \to 0\quad \text{as } n\to\infty. \tag{3}$ The author derives strong convergence of a Palais-Smale sequence in certain cases. In addition, a simple proof of Bethuel’s convergence theorem is also given.
Reviewer: G.Tóth (Camden)

##### MSC:
 5.8e+21 Harmonic maps, etc.