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

The authors study the semilinear elliptic equations \[ -\Delta u = \lambda u |u|^{q-2} + \mu u|u|^{p-2} \] in an open bounded domain \(\Omega \subset \mathbb{R}^N\) with Dirichlet boundary conditions, here \(1 < q < 2 < p < 2N/(N-2)\).
Using variational methods they obtain the remarkable result: for \(\lambda > 0\) and \(\mu \in \mathbb{R}\) arbitrary there exists a sequence \((v_k)\) of solutions with negative energy converging to 0 as \(k \to \infty\). Moreover, for \(\mu > 0\) and \(\lambda \in \mathbb{R}\) 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)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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