Nontrivial solutions for resonance quasilinear elliptic systems

Author:

Borgia Natalino1,Cingolani Silvia1,Vannella Giuseppina2

Affiliation:

1. Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro , Via Orabona 4 , 70125 Bari , Italy

2. Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari , Via Orabona 4 , 70125 Bari , Italy

Abstract

Abstract We establish an Amann-Zehnder-type result for resonance systems of quasilinear elliptic equations with homogeneous Dirichlet boundary conditions, involving nonlinearities growing asymptotically ( p , q ) \left(p,q) -linear at infinity. The proof relies on a cohomological linking in a product Banach space where the properties of cones of the sublevels are missing, differently from the single quasilinear equation. We also perform critical group computations of the energy functional at the origin, in spite of the lack of its C 2 {C}^{2} regularity, to exclude that the found mini-max solution is trivial. Finally, we furnish a local condition which guarantees that the found solution is not semi-trivial.

Publisher

Walter de Gruyter GmbH

Reference48 articles.

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3. L. Boccardo and D. G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 3, 309–323.

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5. A. M. Candela, E. Medeiros, G. Palmieri, and K. Perera, Weak solutions of quasilinear elliptic systems via the cohomological index, Topol. Methods Nonlinear Anal. 36 (2010), no. 1, 1–18.

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