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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
MathSciNet review: 1301008
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Abstract: We study the semilinear elliptic equation $ - \Delta u = \lambda \vert u{\vert^{q - 2}}u + \mu \vert u{\vert^{p - 2}}u$ in an open bounded domain $ \Omega \subset {\mathbb{R}^N}$ with Dirichlet boundary conditions; here $ 1 < q < 2 < p < {2^ \ast }$. Using variational methods we show that 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 $ 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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