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Multiple Solutions to p ‐Laplacian Problems with Asymptotic Nonlinearity as u p −1 at Infinity
Author(s) -
Li Gongbao,
Zhou Huan-Song
Publication year - 2002
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610701002708
Subject(s) - mountain pass theorem , infinity , bounded function , domain (mathematical analysis) , mathematics , p laplacian , type (biology) , nonlinear system , mountain pass , pure mathematics , laplace operator , boundary value problem , mathematical analysis , combinatorics , physics , ecology , quantum mechanics , biology
The paper studies the existence of multiple solutions to the following p ‐Laplacian type elliptic problem ( p > 1):{− Δ p u ( x ) ≡ : div (| ∇ u |p − 2 ∇ u ) = f ( x , u ) x ∈ Ωu ∈ W 0 1 , p( Ω ) ,where Ω is a bounded domain in R N ( N ⩾ 1) with smooth boundary ∂Ω, and f ( x , u ) goes asymptotically in u to ∣ u ∣ p −2 u at infinity. It is well known that this kind of nonlinear term creates some difficulties in the application of the mountain pass theorem because of the lack of an Ambrosetti–Rabinowitz type superlinear condition on f ( x , u ). An improved mountain pass theorem is used to prove that the above problem possesses multiple solutions under some natural conditions on f ( x , u ), and some known results are generalized.