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There are many interesting variational problems for which the Palais-Smale condition cannot be verified. In cases where the Palais-Smale condition can be verified for an approximating integral, and the critical points converge, a Morse theory is valid. This theory applies to a class of variational problems consisting of the energy integral for harmonic maps with a lower order potential.
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Karen Uhlenbeck (1981) studied this question.
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