Estimates for eigenfunctions and eigenvalues of nonlinear elliptic problems

Abstract:

We consider solutions to the nonlinear eigenvalue problem \[ (*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\text {in}} \Omega ,\quad \vec u = 0\:\quad {\text {on}} \partial \Omega ,\quad \vec u{\text { = }}0,\quad {\text {on}}\partial \Omega ,\quad \vec {u} = 0,\] where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and $\Omega \subseteq \mathbf {R}^{n}$ is a smooth bounded domain. We obtain lower bounds for $\lambda$ in the case where $f(x,\vec u)$ has linear growth, and relations between $\lambda ,\Omega$, and ess sup$|\vec u|$ when $f(x,\vec u)$ 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.