Abstract: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.
Keywords: nonlinear boundary value problems, elliptic partial differential equations, bifurcation, resonace
AMS Subject Classification: 35J65, 35B32, 35B34