Oscillating global continua of positive solutions of semilinear elliptic problems

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Abstract

Let O be a bounded domain in Rn, n = 1, with C2 boundary ?O, and consider the semilinear elliptic boundary value problem Lu = ?au + g(·, u)u, in O, u = 0, on ?O, where L is a uniformly elliptic operator on fi, O a ? C0(O¯), a is strictly positive in O¯, and the function g : O¯ × R ? R is continuously differentiable, with g(x, 0) = 0, x ? O¯. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue ?1 of the linear problem. We show that under certain oscillation conditions on the nonlincarity g, this continuum oscillates about ?1, in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each A in an open interval containing ?1. © 1099 American Mathematical Society.

Original languageEnglish
Pages (from-to)229-236
Number of pages8
JournalProceedings of the American Mathematical Society
Volume128
Issue number1
Publication statusPublished - 2000

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Semilinear Elliptic Problem
Positive Solution
Continuum
Semilinear Elliptic Boundary Value Problem
Open interval
Principal Eigenvalue
Continuously differentiable
Strictly positive
G-function
Elliptic Operator
Bounded Domain
Infinity
Oscillation

Keywords

  • Global bifurcation
  • Semilinear elliptic equations

Cite this

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abstract = "Let O be a bounded domain in Rn, n = 1, with C2 boundary ?O, and consider the semilinear elliptic boundary value problem Lu = ?au + g(·, u)u, in O, u = 0, on ?O, where L is a uniformly elliptic operator on fi, O a ? C0(O¯), a is strictly positive in O¯, and the function g : O¯ × R ? R is continuously differentiable, with g(x, 0) = 0, x ? O¯. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue ?1 of the linear problem. We show that under certain oscillation conditions on the nonlincarity g, this continuum oscillates about ?1, in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each A in an open interval containing ?1. {\circledC} 1099 American Mathematical Society.",
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T1 - Oscillating global continua of positive solutions of semilinear elliptic problems

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N2 - Let O be a bounded domain in Rn, n = 1, with C2 boundary ?O, and consider the semilinear elliptic boundary value problem Lu = ?au + g(·, u)u, in O, u = 0, on ?O, where L is a uniformly elliptic operator on fi, O a ? C0(O¯), a is strictly positive in O¯, and the function g : O¯ × R ? R is continuously differentiable, with g(x, 0) = 0, x ? O¯. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue ?1 of the linear problem. We show that under certain oscillation conditions on the nonlincarity g, this continuum oscillates about ?1, in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each A in an open interval containing ?1. © 1099 American Mathematical Society.

AB - Let O be a bounded domain in Rn, n = 1, with C2 boundary ?O, and consider the semilinear elliptic boundary value problem Lu = ?au + g(·, u)u, in O, u = 0, on ?O, where L is a uniformly elliptic operator on fi, O a ? C0(O¯), a is strictly positive in O¯, and the function g : O¯ × R ? R is continuously differentiable, with g(x, 0) = 0, x ? O¯. A well known result of Rabinowitz shows that an unbounded continuum of positive solutions of this problem bifurcates from the principal eigenvalue ?1 of the linear problem. We show that under certain oscillation conditions on the nonlincarity g, this continuum oscillates about ?1, in a certain sense, as it approaches infinity. Hence, in particular, the equation has infinitely many positive solutions for each A in an open interval containing ?1. © 1099 American Mathematical Society.

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