Multiparameter Elliptic Equations in Annular Domains

Abstract

Using fixed point theorems of cone expansion/compression type, the upper-lower solutions method and degree arguments, we study existence, non-existence and multiplicity of positive solutions for a class of second-order ordinary differential equations with multiparameters. We apply our results to semilinear elliptic equations in bounded annular domains with non-homogeneous Dirichlet boundary conditions. More precisely, we apply our main results to equations of the form

$$ \begin{array}{*{20}c} \begin{gathered} - \Delta u \hfill \\ u\left( x \right) \hfill \\ u\left( x \right) \hfill \\ \end{gathered} & \begin{gathered} = \hfill \\ = \hfill \\ = \hfill \\ \end{gathered} & \begin{gathered} \lambda f\left( {\left| x \right|,u} \right) \hfill \\ a \hfill \\ b \hfill \\ \end{gathered} \\ \end{array} \begin{array}{*{20}c} \begin{gathered} in \hfill \\ on \hfill \\ on \hfill \\ \end{gathered} & \begin{gathered} r_1 < \left| x \right| < r_{2,} \hfill \\ \left| x \right| = r_1 , \hfill \\ \left| x \right| = r_2 , \hfill \\ \end{gathered} \\ \end{array} $$

, where a, b and λ are non-negative parameters. One feature of the hypotheses on the nonlinearities that we consider is that they have some sort of local character.

Work partially supported by PADCT/CT-INFRA/CNPq/MCT Grant # 620120/2004-5 and Millennium Institute for the Global Advancement of Brazilian Mathematics — IM-AGIMB, UTA-Grant 4732-04 and FONDECYT Grant # 1040990.