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MATHEMATICA BOHEMICA, Vol. 127, No. 3, pp. 481-496 (2002)
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# 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: ` kucera@math.cas.cz`

**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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