Multiplicity Results for Superlinear Elliptic Equations

Abstract

In this note we will present some recent progress in the question of the existence of multiple solutions of superlinear elliptic boundary value problems. More precisely, we will study equations of the form $$\left\{ {\begin{array}{*{20}{c}} { - \Delta u - f\left( u \right) = h, in \Omega \subset {\mathbb{R}^n}} \\ {u = 0, on \partial \Omega } \end{array}} \right.$$ (1) where Ω is a bounded region in ℝf , and f ∈ C(R) satisfies $$\begin{gathered} {f^ + }: = \mathop {\lim }\limits_{s \to + \infty } \inf \frac{{f\left( s \right)}}{s} = + \infty \hfill \\ {f^ - }: = \mathop {\lim }\limits_{s \to - \infty } \sup \frac{{f\left( s \right)}}{s} < + \infty \hfill \\ \end{gathered} $$ (2) We are interested to find (estimates for) the number of solutions that exist for equation (1) in dependence of a given function h.