Opuscula Math. 36, no. 1 (2016), 81-101

Opuscula Mathematica

Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces

Ionela-Loredana Stăncuţ
Iulia Dorotheea Stîrcu

Abstract. In this paper we consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential \(V\) on a bounded domain in \(\mathbb{R}^N\) (\(N\geq 3\)) with a smooth boundary. We establish three main results with various assumptions. The first one asserts that any \(\lambda\gt 0\) is an eigenvalue of our problem. The second theorem states the existence of a constant \(\lambda_{*}\gt 0\) such that any \(\lambda\in(0,\lambda_{*}]\) is an eigenvalue, while the third theorem claims the existence of a constant \(\lambda^{*}\gt 0\) such that every \(\lambda\in[\lambda^{*}, \infty)\) is an eigenvalue of the problem.

Keywords: anisotropic Orlicz-Sobolev space, potential, critical point, weak solution, eigenvalue.

Mathematics Subject Classification: 35D30, 35J60, 58E05.

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  • Ionela-Loredana Stăncuţ
  • University of Craiova, Department of Mathematics, 200585 Craiova, Romania
  • Iulia Dorotheea Stîrcu
  • University of Craiova, Department of Mathematics, 200585 Craiova, Romania
  • Communicated by Vicentiu D. Radulescu.
  • Received: 2015-01-09.
  • Accepted: 2015-03-23.
  • Published online: 2015-09-19.
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Cite this article as:
Ionela-Loredana Stăncuţ, Iulia Dorotheea Stîrcu, Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces, Opuscula Math. 36, no. 1 (2016), 81-101, http://dx.doi.org/10.7494/OpMath.2016.36.1.81

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