Max-min Characterization of the Mountain Pass Energy Level for a Class of Variational Problems

We provide a max-min characterization of the mountain pass energy level for a family of variational problems. As a consequence we deduce the mountain pass structure of solutions to suitable PDEs, whose existence follows from classical minimization argument. In the literature the existence of solutions for variational PDE is often reduced to the existence of critical points of functionals F having the following structure F (u) = T (u)− U(u) with u belonging to a Banach space X and T, U that satisfy suitable conditions. A classical strategy which is very useful in order to find critical points for the functional F is to look at the following minimization problem min U(u)=1 T (u) and eventually to remove the Lagrange multiplier that appears by using suitable invariances of the problem. On the other hand when the functional F shows a mountain pass geometry it is customary to look at critical points of the unconstrained functional F (u) on the whole space X . More specifically it is well known that a candidate critical value is given by the mountain pass energy level (0.1) c := infη∈Γsupt∈[0,1]F (η(t)) where (0.2) Γ := {η ∈ C([0, 1];X |η(0) = 0, F (η(1)) < 0}. A natural question is to understand whether or not the two forementioned approaches provide the same solution. Of course in both approaches described above the main difficulty is related with the eventual lack of compactness respectively for the minimizing sequences and for the Palais Smale sequences. The main contribution of this paper is to give a max-min characterization of the mountain pass value c described above under a general framework. Roughly speaking we relate the existence of a regular path of minimizers for min U(u)=λ T (u), λ ∈ R with the mountain pass energy level. As a byproduct we shall show the mountain pass structure of solutions obtained via minimization approach to a family of PDEs with subcritical and critical Sobolev exponent. Finally we mention the papers [3] and [5] where the question mentioned above is studied for specific PDEs. We shall recover those results as a consequence of our general topological result stated below. 1 2 JACOPO BELLAZZINI AND NICOLA VISCIGLIA In order to give a concrete example in which for instance it is not easy to reveal the mountain pass structure of solutions obtained via minimization, we consider the following equation: (0.3) −∆pu = |u| p∗−2u on R with 1 < p < n and p = np n− p It is well known since [6] that the best constant is achieved in the following continuous embedding D(R) ⊂ L ∗ (R) and hence via rescaling argument there is a nontrivial solution of (0.3). As a consequence of our next abstract theorem one can deduce easily that any solution constructed as above via a minimization procedure is a mountain pass solution. We fix some notations. For every λ ∈ R we introduce the sets Uλ := {u ∈ X |U(u) = λ} and iλ := inf u∈Uλ T (u). Finally we introduce Mλ := {u ∈ Uλ|T (u) = iλ} (notice that it could even be the empty set). Theorem 0.1. Let X be a Banach space and F = T − U with (0.4) T, U ∈ C(X,R), T (u) ≥ 0, T (0) = U(0) = 0, (0.5) lim λ→0+ iλ = 0 (0.6) λ > 0 where λ := inf{λ ∈ (0,∞) such that iλ − λ ≤ 0} (0.7) λ > 0 where λ := inf{λ ∈ (0,∞) such that iλ − λ < 0} (0.8) Mλ 6= ∅ ∀λ ∈ R + there exists a continuous map γ : [0,∞) → X (0.9) such that γ(λ) ∈ Mλ. Then c > 0 and