Advances in Differential Equations

Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states

H. Jeanjean and C. A. Stuart

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For a semilinear second order differential equation on $\mathbb{R}$, the existence of a continuous curve of positive solutions which bifurcates from the lowest eigenvalue of the linearized problem is proved. This curve can be parameterized globally by $\lambda$ and can be extended to infinity. We establish that all solutions of the equation are even and monotone, and under appropriate conditions, all of them belong to the curve of bifurcation. Our results depend heavily on the combination of symmetry and monotonicity imposed on the equation.

Article information

Adv. Differential Equations, Volume 4, Number 5 (1999), 639-670.

First available in Project Euclid: 15 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34B15: Nonlinear boundary value problems
Secondary: 47J15: Abstract bifurcation theory [See also 34C23, 37Gxx, 58E07, 58E09] 58E07: Abstract bifurcation theory 78A50: Antennas, wave-guides 78A60: Lasers, masers, optical bistability, nonlinear optics [See also 81V80]


Jeanjean, H.; Stuart, C. A. Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states. Adv. Differential Equations 4 (1999), no. 5, 639--670.

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