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.


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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