Global branching for discontinuous problems

Antonio Ambrosetti; Recalde Marco Calahorrano; Fernando R. Dobarro

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We deal with the boundary value problem $\begin{matrix} - Δ u (x) & = λ_{1} u (x) + g (\nabla u (x)) + h (x), & x \in Ω u (x) & = 0, & x \in \partial Ω \end{matrix}$ where $Ω \subset ℝ^{N}$ is an smooth bounded domain, $λ_{1}$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $Ω$ , $h \in L^{max {2, N / 2}} (Ω)$ and $g : ℝ^{N} ⟶ ℝ$ is bounded and continuous. Bifurcation theory is used as the right framework to show the existence of solution provided that $g$ satisfies certain conditions on the origin and at infinity.

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