Model Elliptic Problems

Abstract

In Chapter I, we study the problem

$$ \left. {\begin{array}{*{20}c} { - \Delta u = f\left( {x,u} \right),{\mathbf{ }}x \in \Omega } \\ {u = 0,{\mathbf{ }}x \in \partial \Omega ,} \\ \end{array} {\mathbf{ }}} \right\} $$

where f: Ω × ℝ → ℝ is a Carathéodory function (i.e. f(·, u) is measurable for any u ∈ ℝ and f(x, ·) is continuous for a.e. x ∈ Ω). Of course, the boundary condition in (2.1) is not present if ω = ℝⁿ. We will be mainly interested in the model case

$$ f\left( {x,u} \right) = |u|^{p - 1} u + \lambda u,{\mathbf{ }}where{\mathbf{ }}p > 1{\mathbf{ }}and{\mathbf{ }}\lambda {\mathbf{ }} \in \mathbb{R} $$

Denote by p_s the critical Sobolev exponent,

$$ ps: = \left\{ {\begin{array}{*{20}c} {\infty {\mathbf{ }}if{\mathbf{ }}n \leqslant 2,} \\ {\left( {n + 2} \right)/\left( {n - 2} \right){\mathbf{ }}if{\mathbf{ }}n > 2,} \\ \end{array} } \right. $$

We shall refer to the cases p p_s, p = p_s or p p_s as to (Sobolev) subcritical, critical or supercritical, respectively.