### Abstract

We consider the boundary value problem(Pλ)u∈H01(Ω)∩L∞(Ω):-δu=λc(x)u+μ(x)|∇u|2+h(x), where Ω⊂RN, N≥3 is a bounded domain with smooth boundary. It is assumed that c{greater-than but not equal to}0, c, h belong to L^{p}(Ω) for some p>N/2 and that μ∈L^{∞}(Ω). We explicitly describe a condition which guarantees the existence of a unique solution of (P_{λ}) when λ<0 and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of (P_{0}). It crosses the axis λ=0 if (P_{0}) has a solution, otherwise it bifurcates from infinity at the left of the axis λ=0. Assuming that (P_{0}) has a solution and strengthening our assumptions to μ(x)≥μ_{1}>0 and h{greater-than but not equal to}0, we show that the continuum bifurcates from infinity on the right of the axis λ=0 and this implies, in particular, the existence of two solutions for any λ>0 sufficiently small.

Original language | English |
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Pages (from-to) | 2298-2335 |

Number of pages | 38 |

Journal | Journal of Functional Analysis |

Volume | 268 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2015 Apr 15 |

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### Keywords

- Continuum of solutions
- Elliptic equations
- Quadratic growth in the gradient
- Topological degree

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*268*(8), 2298-2335. https://doi.org/10.1016/j.jfa.2015.01.014