Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Abstract

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter γ ∈ (0, 1] corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for γ = 1. Thus, by analyzing the case γ = 1 we emphasize specific properties of the physically relevant parameter γ in the vortex concentration phenomena.2000 Mathematics Subject Classification. 35J91, 35A01, 35B44, 35B30.Theorem 1.1. Fix γ ∈ (0, 1]. Assume that Ω satisifes the symmetry assumption (1.9). For any k ∈ N there exists ρ 0 > 0 such that for all ρ ∈ (0, ρ 0 ) problem (1.1) admits a concentrating sign-changing family of solutions u ρ ∈ H γ satisfying ρe uρ dx * ⇀ m + (0) δ 0 (dx), ρτ e −γuρ dx * ⇀ m − (0) δ 0 (dx) as ρ → 0 +

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Uniform bounds for solutions to elliptic problems on simply connected planar domains

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Sign-changing solutions for the one-dimensional non-local sinh-Poisson equation

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