MATHEMATICA BOHEMICA, Vol. 127, No. 3, pp. 481-496 (2002)

Bifurcations for a problem with jumping nonlinearities

Lucie Karna, Milan Kucera

L. Karna, Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, Technicka 2, Praha 6, Czech Republic
M. Kucera, Mathematical Institute of the Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic, e-mail:

Abstract: A bifurcation problem for the equation $$ \Delta u+\lambda u-\alpha u^++\beta u^-+g(\lambda ,u)=0 $$ in a bounded domain in $\R ^N$ with mixed boundary conditions, given nonnegative functions $\alpha ,\beta \in L_\infty $ and a small perturbation $g$ is considered. The existence of a global bifurcation between two given simple eigenvalues $\lambda ^{(1)},\lambda ^{(2)}$ of the Laplacian is proved under some assumptions about the supports of the functions $\alpha ,\beta $. These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to $\lambda ^{(1)}, \lambda ^{(2)}$.

Keywords: nonlinearizable elliptic equations, jumping nonlinearities, global bifurcation, half-eigenvalue

Classification (MSC2000): 35B32, 35J65

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