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

Arcoya, David; del Toro, Naira

Commentat. Math. Univ. Carol. 44, No. 3, 413-426 (2003).

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

Keywords:

nonlinear boundary value problem; elliptic partial differential equations; bifurcation; resonance