Estimates for eigenfunctions and eigenvalues of nonlinear elliptic problems

Abstract

We consider solutions to the nonlinear eigenvalue problem \[ ( ∗ ) A ( x , u → ) u → + λ f ( x , u → ) = 0 in Ω , u → = 0 on ∂ Ω , u → = 0 , on ∂ Ω , u → = 0 , (*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\text {in}}\,\Omega ,\quad \vec u = 0\:\quad {\text {on}}\,\partial \Omega ,\quad \vec u{\text { = }}0,\quad {\text {on}}\partial \Omega ,\quad \vec {u} = 0, \] where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and Ω ⊆ R n \Omega \subseteq \mathbf {R}^{n} is a smooth bounded domain. We obtain lower bounds for λ \lambda in the case where f ( x , u → ) f(x,\vec u) has linear growth, and relations between λ , Ω \lambda ,\Omega , and ess sup | u → | |\vec u| when f ( x , u → ) f(x,\vec u) has sub- or superlinear growth. The estimates are based on integration by parts and application of certain Sobolev inequalities. We briefly discuss extensions to higher order systems.