Concentration Compactness at the Mountain Pass Level in Semilinear Elliptic Problems

Abstract

Let c be the usual mountain pass level for the semilinear elliptic functional $$G(u) = \frac{1}{2} \int_{\Omega} (\mid\nabla u\mid^{2} + \lambda u^{2})dx - \int_{\Omega} F(x, u(x))dx. \quad (0.1)$$ . In general, c ≤ c#, where c# is the analogous mountain pass level of the asymptotic functional G# defined with respect to unbounded shifts or dilations. We show under general conditions that whenever the strict inequality c ≤ c# holds, the functional G satisfies the Palais – Smale condition at the level c and, consequently, has a critical point at this level. This sets a solvability framework for unconstrained mountain pass similar to that of P.-L.Lions set for constrained minimization. The nonlinearity F is allowed to have critical growth with asymptotically selfsimilar oscillations about the critical “stem” \(|u|^{2*}\) and not only “stem” asymptotics. For example, the main existence result, Theorem 3.2, holds for \(F(x, s) = |s|^{2*} e^ {\frac {\sigma N x^{2}}{1+x^{2}} {\rm sin}(2 log(|s|))}\). Since the unconstrained minimax is studied, the convexity-type conditions that arise with the use of Nehari constraint \((G'(u), u) = 0\) are not required.