TY - JOUR

T1 - Multiplicity Result for a Degenerate Elliptic Equation with Logistic Reaction

AU - Takeuchi, Shingo

PY - 2001/6/10

Y1 - 2001/6/10

N2 - A degenerate elliptic equation λΔpu+uq-1(1-ur)=0 with zero Dirichlet boundary condition, where λ is a positive parameter, 20, is studied. It has been known that there exists a positive number Λ such that if λ>Λ, then the problem has no positive solution; if λ≤Λ, then there exists a maximal solution; and for sufficiently small λ (<Λ), the problem has at least two solutions. In this article, it is shown that if λ<Λ, then there exist at least two positive solutions, via the variational method.

AB - A degenerate elliptic equation λΔpu+uq-1(1-ur)=0 with zero Dirichlet boundary condition, where λ is a positive parameter, 20, is studied. It has been known that there exists a positive number Λ such that if λ>Λ, then the problem has no positive solution; if λ≤Λ, then there exists a maximal solution; and for sufficiently small λ (<Λ), the problem has at least two solutions. In this article, it is shown that if λ<Λ, then there exist at least two positive solutions, via the variational method.

KW - Degenerate elliptic equation

KW - Mountain pass theorem

KW - Multiple solutions

KW - P-Laplace operator

UR - http://www.scopus.com/inward/record.url?scp=0042835751&partnerID=8YFLogxK

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U2 - 10.1006/jdeq.2000.3914

DO - 10.1006/jdeq.2000.3914

M3 - Article

AN - SCOPUS:0042835751

VL - 173

SP - 138

EP - 144

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 1

ER -