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Model Elliptic Problems

Part of the Birkhäuser Advanced Texts / Basler Lehrbücher book series (BAT)

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 ps 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 ps, p = ps or p ps as to (Sobolev) subcritical, critical or supercritical, respectively.

Keywords

Weak Solution Maximum Principle Classical Solution Bifurcation Diagram Variational Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2007

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