Bifurcation for an elliptic problem with nonlinear boundary conditions
Published 2012-11-28
Keywords
- Bifurcation from infinity,
- stability,
- instability,
- multiplicity,
- resonance
- turning points ...More
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Abstract
This paper gives a survey over bifurcation problems for elliptic equations with nonlinear boundary conditions depending on a real parameter. We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, and characterize when they are sub or supercritical. Furthermore, we apply these results and techniques to obtain Landesman-Lazer type conditions guarantying the existence of solutions in the resonant case and to obtain a uniform Anti-Maximum Principle and several results related to the spectral behavior when the potential at the boundary is perturbed. We also characterize the stability type of the solutions in the unbounded branches. In the remainder of this paper, we start our analysis on a sublinear oscillatory nonlinearity. We first focus our attention on the loss of Landesman-Lazer type conditions, and even in that situation, we are able to prove the existence of infinitely many resonant solutions and infinitely many turning points. Next we focus our attention on stability switches. Even in the absence of resonant solutions, we are able to provide sufficient conditions for the existence of sequences of stable solutions, unstable solutions, and turning points. We also discuss on bifurcation from the trivial solution set, and on a sublinear oscillatory nonlinearity. Finally, we states a formula for the derivative of a localized Steklov eigenvalue on a subset of the boundary, with respect to tangential variations of that subset.
Keywords: Bifurcation from infinity, stability, instability, multiplicity, resonance, turning points.
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References
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