Multiple solutions for resonant semilinear elliptic problems in [formula omitted]

Abstract

We prove the existence of multiple nontrivial solutions for the semilinear elliptic problem − Δ u = h ( λ u + g ( u ) ) in R N , u ∈ D 1 , 2 , where h ∈ L 1 ∩ L α for α > N / 2 , N ⩾ 3 , g is a C 1 ( R , R ) function that has at most linear growth at infinity, g ( 0 ) = 0 , and λ is an eigenvalue of the corresponding linear problem − Δ u = λ h u in R N , u ∈ D 1 , 2 . Existence of multiple solutions, for certain values of g ′ ( 0 ) , is obtained by imposing a generalized Landesman–Lazer type condition. We use the saddle point theorem of Ambrosetti and Rabinowitz and the mountain pass theorem, as well as a Morse-index result of Ambrosetti [A. Ambrosetti, Differential Equations with Multiple Solutions and Nonlinear Functional Analysis, Equadiff 82, Lecture Notes in Math., vol. 1017, Springer-Verlag, Berlin, 1983] and a Leray–Schauder index theorem for mountain pass type critical points due to Hofer [H. Hofer, A note on the Topological Degree at a critical Point of Mountain Pass Type, Proc. Amer. Math. Soc. 90 (1984) 309–315]. The results of this paper are based upon multiplicity results for resonant problems on bounded domains in [E. Landesman, S. Robinson, A. Rumbos, Multiple solutions of semilinear elliptic problems at resonance, Nonlinear Anal. 24 (1995) 1049–1059] and [S. Robinson, Multiple solutions for semilinear elliptic boundary value problems at resonance, Electron. J. Differential Equations 1995 (1995) 1–14], and complement a previous existence result by the authors in [G. López Garza, A. Rumbos, Resonance and strong resonance for semilinear elliptic equations in R N , Electron. J. Differential Equations 2003 (2003) 1–22] for resonant problems in R N in which g was assumed to be bounded.