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An eigenvalue problem for a quasilinear elliptic equation
Luleå University of Technology.
1998 (English)In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 196, no 1, p. 43-59Article in journal (Refereed) Published
Abstract [en]

In this paper, we are concerned with the following eigenvalue problem: {-Δpu = λf(x, u) in Ω, u = 0 on ∂Ω, here Ω ⊂ ℝn (n ≥ 2) is a C1,α - domain and - Δp is the degenerate p-Laplace operator with p > 1. An interesting special case is when f = φ(x)\u\q1-1u-ψ(x)\u\q2-1u, 0 < q1 < q2. By using the sub- and supersolutions method and the variational method, we prove the existence of the solution of the problem under certain growth conditions for f(x, u). In the above special case, our results give the existence of at least one positive solution for q2 > p* - 1, where p* is the critical Sobolev exponent, and the existence of at least two positive and two negative solutions for q1 < p - 1 < q2 < p* - 1. We also present a 1D example which has many positive solutions for certain interval of λ and for a special value of λ it has even infinitely many solutions.

Place, publisher, year, edition, pages
1998. Vol. 196, no 1, p. 43-59
Keywords [en]
Critical points, Multiplicity, Nonlinearity, Sub- and supersolutions
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-4747DOI: 10.1002/mana.19981960103ISI: 000077775200002Scopus ID: 2-s2.0-0032435920Local ID: 2bcf8d50-b07b-11db-840a-000ea68e967bOAI: oai:DiVA.org:ltu-4747DiVA, id: diva2:977621
Note

Godkänd; 1998; 20070130 (kani)

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2022-05-04Bibliographically approved

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