Using the 'monotonicity trick' introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais-Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the formWe assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity / satisfies (i) f(x, s)s~1 -¥ a 6]0, oo) as s ->• +oo; and (ii) f(x, s)s~x is non decreasing as a function of s ^ 0, a.e. x E M. N .

No abstract

No abstract

We consider singularly perturbed elliptic equations ε 2 ∆u−V (x)u+ f (u) = 0, x ∈ R N , N ≥ 3. For small ε > 0, we glue together localized bound state solutions concentrating at isolated components of positive local minimum of V under conditions on f we believe to be almost optimal.