On an elliptic equation with concave and convex nonlinearities

Authors:
Thomas Bartsch and Michel Willem

Journal:
Proc. Amer. Math. Soc. **123** (1995), 3555-3561

MSC:
Primary 35J65; Secondary 58E05

DOI:
https://doi.org/10.1090/S0002-9939-1995-1301008-2

MathSciNet review:
1301008

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Abstract: We study the semilinear elliptic equation in an open bounded domain with Dirichlet boundary conditions; here . Using variational methods we show that for and arbitrary there exists a sequence of solutions with negative energy converging to 0 as . Moreover, for and arbitrary there exists a sequence of solutions with unbounded energy. This answers a question of Ambrosetti, Brézis and Cerami. The main ingredient is a new critical point theorem, which guarantees the existence of infinitely many critical values of an even functional in a bounded range. We can also treat strongly indefinite functionals and obtain similar results for first-order Hamiltonian systems.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1301008-2

Article copyright:
© Copyright 1995
American Mathematical Society