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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.

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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