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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)

58E20 Harmonic maps, etc.