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We improve the Poincaré inequality, the Sobolev embedding theorem and the Trudinger embedding theorem and prove a mountain‐pass theorem. Applying these results we study a nonlinear singular mixed boundary problem.

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Abstract. Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the C 1 -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.

We get the existence and regularity of minimizers of certain functionals, which may be degenerate and have non-polynomial growth. Applying these results we can find exponentially harmonic maps in every homotopy class of maps from a connected Riemannian C4-manifold into [Formula: see text], where [Formula: see text] is a compact Riemannian C4-manifold.