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On an elliptic equation with concave and convex nonlinearities. (English) Zbl 0848.35039

The authors study the semilinear elliptic equations

-Δu=λu|u| q-2 +μu|u| p-2

in an open bounded domain Ω N with Dirichlet boundary conditions, here 1<q<2<p<2N/(N-2).

Using variational methods they obtain the remarkable result: for λ>0 and μ arbitrary there exists a sequence (v k ) of solutions with negative energy converging to 0 as k. Moreover, for μ>0 and λ arbitrary there exists a sequence of solutions with unbounded energy. A similar result is obtained for first order Hamiltonian systems. The main ingredient in the proofs is a new critical point theorem, which guarantees the existence of infinitely many critical values of a functional with symmetries in a bounded range.

Reviewer: V.Moroz (Minsk)

35J65Nonlinear boundary value problems for linear elliptic equations
34C25Periodic solutions of ODE
58E05Abstract critical point theory