Mountain pass on a closed convex set (Q1353726)

The author proves a version of the mountain pass theorem for a functional defined on a closed convex set of a Banach space. Such a functional does not necessarily satisfy the Palais-Smale condition. This abstract result is applied to the study of the semilinear Dirichlet problem \[ \begin{cases} -\Delta u+u^2 =f(x) \quad & \text{in } \Omega,\\ u=0 \quad & \text{on } \partial \Omega,\end{cases} \tag{*} \] where \(\Omega\) is a piecewise smooth and bounded domain in \(\mathbb{R}^n\), \(n<6\). The problem (*) was studied by K. C. Chang and J. Nirenberg who proved the existence of a negative and a nontrivial solution for (*). In this paper the author shows the existence of a positive solution for (*).