A Variant of the Mountain Pass Theorem and Variational Gluing
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ilan Journal of Mathematics
A Variant of the Mountain Pass Theorem and Variational Gluing Piero Montecchiari and Paul H. Rabinowitz Abstract. This paper surveys some recent work on a variant of the Mountain Pass Theorem that is applicable to some classes of differential equations involving unbounded spatial or temporal domains. In particular its application to a system of semilinear elliptic PDEs on Rn and to a family of Hamiltonian systems involving double well potentials will also be discussed. Mathematics Subject Classification (2010). Primary: 35J50, 35J47; Secondary: 35J57, 34C37. Keywords. Variational methods, mountain pass theorem, variational gluing, nondegeneracy condition, heteroclinic solutions, homoclinic solutions, double well potential, multitransition solutions.
1. Introduction The Mountain Pass Theorem is a useful tool for obtaining the existence of solutions of differential equations that arise as critical points of a functional, I, defined on a Banach space, E, or a subset thereof. The differential equations in their weak form correspond to I (u) = 0, I denoting the Frechet derivative of I. The functional is required to possess an appropriate geometric structure. The simplest example of that structure occurs when I has a local minimum that is not a global minimum. The other requirements of the Theorem are some smoothness for I, e.g. I ∈ C 1 (E, R), and some compactness such as is embodied in the Palais-Smale condition, (P S): any sequence for which I is bounded and I → 0 possesses a convergent subsequence. The Mountain Pass Theorem also provides a minimax characterization of the associated critical value, c: (1.1) c = inf max I(g(θ)) g∈Γ θ∈[0,1]
where Γ = {g ∈ C([0, 1], E) | g(0) = 0 and g(1) = e}
and the local minimum of I occurs at 0 with I(0) = 0 and e = 0 is such that I(e) ≤ 0.
2
P. Montecchiari Montecchiari and and P.H. P.H. Rabinowitz Rabinowitz P.
In most applications, E is a class of functions having a bounded temporal or spatial domain. However if the domain is unbounded, even if the smoothness and geometric conditions are satisfied, the (P S) condition may fail. Two examples in which this happens will be considered in Section 2. The first example involves a family of semilinear elliptic equations on Rn of the form: −∆u + u = Fu (x, u),
a(x)|u|p−1 u
x ∈ Rn .
(PDE)
n+2 n−2
A model case here is F = where 1 < p < with n > 2 or 1 < p < +∞ when n = 1, 2, and a(x) is periodic in the each of the components of x. The second example is a family of second order Hamiltonian systems with a double well potential on Rm : (HS) −¨ q + Vq (t, q) = 0, t ∈ R, q ∈ Rm .
where V is 1-periodic in t. Geometrically in Example 1, the associated functional, I, has 0 as a local but not global minimum while in Example 2, I has a pair of global minima at the minima of the potential. Moreover due to the periodicity of a and V , the associated functional, I is invariant under a translational symmetry. This additional information together with the aid of concentration compactness arguments allows one
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