Geodesic
Semilinear elliptic problems with nonlinearities depending on the derivative
Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 3, pp. 413-426.

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We deal with the boundary value problem $$ \alignat2 -\Delta u(x) = \lambda _{1}u(x)+g(\nabla u(x))+h(x), \quad x\in \Omega \ u(x) = 0, x\in \partial \Omega \endalignat $$ where $\Omega \subset \Bbb R^N$ is an smooth bounded domain, $\lambda _{1}$ is the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions on $\Omega $, $h\in L^{\max \{2,N/2\}}(\Omega )$ and $g:\Bbb R^N\longrightarrow \Bbb R$ 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.
Classification : 35B32, 35B34, 35J25, 35J60, 35J65, 47J15
Keywords: nonlinear boundary value problems; elliptic partial differential equations; bifurcation; resonace 
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     title = {Semilinear elliptic problems with nonlinearities depending on the derivative},
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Arcoya, David; del Toro, Naira. Semilinear elliptic problems with nonlinearities depending on the derivative. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) no. 3, pp. 413-426. http://geodesic.mathdoc.fr/item/CMUC_2003__44_3_a2/