An elliptic equation with no monotonicity condition on the nonlinearity

Spradlin, Gregory S.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 786-794

An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree theory argument must be used.

Classification : 35J20, 35J60
Keywords: mountain-pass theorem, variational methods, Nehari manifold, Brouwer degree, concentration-compactness

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