Classical Variational Method
I. A. Kuzin, Stanislav I. Pohožaev · Birkhäuser Basel eBooks · 1997
In this chapter and throughout the book, we shall consider boundary problems of the form $$ - \Delta u + f({\text{x}},u) = 0\quad {\text{on}}\;{\mathbb{R}^N}, u \to 0\quad \;{\text{as}}\;\left| {\text{x}} \right| \to \infty.$$ (1.1) Functions f : ℝ N x ℝ → ℝ are supposed to satisfy a so-called Carathéodory condition, i.e., to be continuous with respect to t for almost all x ∈ ℝN and measurable in x for all t ∈ ℝ.