Flows and Critical Points

Abstract

In this chapter we study equations of the form

$$\displaystyle \begin{aligned} \left\{\begin{aligned} - {\Delta_{\mathit p}} u & = f(x,u)\;\quad && \;\text{in }\; \Omega\\ {} u & = 0 && \text{on }\; \partial{\Omega} \end{aligned}\right \}, \end{aligned}$$

where Ω is a bounded domain in $\mathbb R^n,\, n \ge 1$, ${\Delta _{\mathit p}} u = \operatorname {\mathrm {div}} \big (|\nabla u|{ }^{p-2}\, \nabla u\big )$ is the p-Laplacian of u, 1 < p < ∞, and f is a Carathéodory function on $\Omega \times \mathbb R$ with subcritical growth. We show that sandwich pairs can be used in solving such problems.