## Semilinear elliptic problems with nonlinearities depending on the derivative.(English)Zbl 1105.35038

The paper deals with a semilinear elliptic equation $-\Delta u=\lambda _1u+g(\nabla u)+h\qquad x\in \Omega$ in a smooth domain $$\Omega$$ in $${\mathbb R}^N$$ with boundary condition $$u=0$$ at the boundary $$\partial \Omega$$. For the case of differentiable $$g(\xi )$$ satisfying $$g(0)=0$$, $$\nabla g(0)\neq 0$$ and $$g(\xi )\to 0$$ for $$| \xi | \to \infty$$, the bifurcation problem around the first eigenvalue $$\lambda _1$$ is studied. By means of topological degree technique the following results are proved: for “small” $$h$$ the problem admits at least three solutions while for “big” $$h$$ there is no solution.
Reviewer: Jan Franců (Brno)

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B32 Bifurcations in context of PDEs 35B34 Resonance in context of PDEs
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