We are interested in the existence of standing waves for the nonlinear Klein Gordon equation ∈2□ ψ + W′(ψ) = 0 in a bounded domain D. A standing wave has the form ψ(t,x) = u(x)e-iwt/∈; for these solutions the Klein Gordon equation becomes { -∈2δu + W′(u) = w2u x ∈ D (u,w)∈H10(D) × ℝ. We want to use a Benci-Cerami type argument in order to prove a the existence of several standing waves localized in suitable points of D. The main result of this paper is that, under suitable growth condition on W, for ∈ sufficiently small, we have at least cat(D) stationary solutions of equation (†), where cat(D) is the Ljusternik-Schnirelmann category. The proof is achieved by solving a constrained critical point problem via variational techniques.
Ghimenti, M., Grisanti, C. (2009). Semiclassical limit for the nonlinear Klein Gordon equation. ADVANCED NONLINEAR STUDIES, 9(1), 137-147.
Semiclassical limit for the nonlinear Klein Gordon equation
GHIMENTI, MARCO GIPO;
2009
Abstract
We are interested in the existence of standing waves for the nonlinear Klein Gordon equation ∈2□ ψ + W′(ψ) = 0 in a bounded domain D. A standing wave has the form ψ(t,x) = u(x)e-iwt/∈; for these solutions the Klein Gordon equation becomes { -∈2δu + W′(u) = w2u x ∈ D (u,w)∈H10(D) × ℝ. We want to use a Benci-Cerami type argument in order to prove a the existence of several standing waves localized in suitable points of D. The main result of this paper is that, under suitable growth condition on W, for ∈ sufficiently small, we have at least cat(D) stationary solutions of equation (†), where cat(D) is the Ljusternik-Schnirelmann category. The proof is achieved by solving a constrained critical point problem via variational techniques.
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