# Oscillating global continua of positive solutions of semilinear elliptic problems

Research output: Contribution to journalArticle

### 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 language English 229-236 8 Proceedings of the American Mathematical Society 128 1 Published - 2000

### Fingerprint

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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title = "Oscillating global continua of positive solutions of semilinear elliptic problems",
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.",
keywords = "Global bifurcation, Semilinear elliptic equations",
author = "Rynne, {Bryan P.}",
year = "2000",
language = "English",
volume = "128",
pages = "229--236",
journal = "Proceedings of the American Mathematical Society",
issn = "0002-9939",
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In: Proceedings of the American Mathematical Society, Vol. 128, No. 1, 2000, p. 229-236.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Oscillating global continua of positive solutions of semilinear elliptic problems

AU - Rynne, Bryan P.

PY - 2000

Y1 - 2000

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.

KW - Global bifurcation

KW - Semilinear elliptic equations

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