Bubbling solutions for Moser-Trudinger type equations on compact Riemann surfaces

2017/09/04 by Figueroa, Pablo, Musso, Monica
#35J08 #35J15 #53C20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.01106

Abstract

We study an elliptic equation related to the Moser-Trudinger inequality on a compact Riemann surface (S,g), Δ_g u+λ\Biggl(ue^u2-1\over |S| ∫_S ue^u2 dv_g\Biggl)=0,\quadin S, ∫_S u dv_g=0, where λ>0 is a small parameter, |S| is the area of S, Δ_g is the Laplace-Beltrami operator and dv_g is the area element. Given any integer k≥ 1, under general conditions on S we find a bubbling solution u_λ which blows up at exactly k points in S, as λ→0. When S is a flat two-torus in rectangular form, we find that either seven or nine families of such solutions do exist for k=2. In particular, in any square flat two-torus actually nine families of bubbling solutions with two bubbling points do exist. If S is a Riemann surface with non-constant Robin's function then at least two bubbling solutions with k=1 exists.

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