On a class of parametric Neumann problems with indefinite and unbounded potential

Abstract

We consider a parametric Neumann problem with an indefinite and unbounded potential. Using a combination of critical point theory with truncation and comparison techniques, with Morse theory and with invariance arguments for a suitable negative gradient flow, we prove two multiplicity theorems for certain values of the parameter. In the first theorem we produce three solutions and in the second five. For all solutions we provide sign information. Our work improves significantly results existing in the literature.

Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential

“…Problems with double resonance (that is, possible resonance at both ends of the spectral interval) were studied by O'Regan, Papageorgiou & Smyrlis [25], with β ≡ 0 (see also Hu & Papageorgiou [16] for Dirichlet problems with β = 0). Neumann equations with unbounded, indefinite potential were studied by Papageorgiou & Rȃdulescu [27] (problems with crossing nonlinearity) and Papageorgiou & Smyrlis [28] (coercive problems). Our aim is to prove multiplicity theorems for these problems.…”

Robin problems with indefinite, unbounded potential and reaction of arbitrary growth

“…The extension to Neumann problems (that is, β ≡ 0) with a potential term, was proved by Papageorgiou and Smyrlis [25]. The extension to p-Laplacian equations with ξ ≡ 0 and Robin boundary condition, can be found in the recent work of Papageorgiou and Rȃdulescu [23].…”

“…This eigenvalue problem was studied for the Neumann boundary condition (that is, β ≡ 0), in Papageorgiou and Rȃdulescu [22] and Papageorgiou and Smyrlis [25]. For the p-Laplacian and Neumann boundary condition, it was investigated by Mugnai and Papageorgiou [18] and for the p-Laplacian with Robin boundary condition and ξ ≡ 0 by Papageorgiou and Rȃdulescu [23].…”

“…Using (4) and the spectral theorem for compact self-adjoint operators, exactly as in [22,25], we have a sequence {λ k (β)} k 1 such thatλ k (β) → +∞ as k → ∞ which are all the eigenvalues of (2). Let E(λ k (β)) be the corresponding eigenspace.…”

“…We mention the works of Kyritsi and Papageorgiou [13], Papageorgiou and Papalini [21] (Dirichlet problems), and Papageorgiou and Rȃdulescu [22,24], Papageorgiou and Smyrlis [25] (Neumann problems).…”