Asymptotic Behavior of a Fourth Order Mean Field Equation with Dirichlet Boundary Condition
نویسنده
چکیده
where Σ is a smooth bounded domain in R, has been extensively studied by many authors. Let (uk, ρk) be a unbounded sequence of solutions to (1.2) with ρk ≤ C,maxx∈Σ uk(x) → +∞. Then it has been proved that (P1) (no boundary bubbles) uk is uniformly bounded near a neighborhood of ∂Σ (Nagasaki-Suzuki [33], Ma-Wei [29]); (P2) (bubbles are simple) ρk → 8mπ for some m ≥ 1 and uk(x) → 8π ∑m j=1 G(·, xj) in C loc(Σ\{x1, ..., xm}) (Brézis-Merle [8], Li-Shafrir [24], Nagasaki-Suzuki [33], Ma-Wei [29]), where G is the Green function of −∆ with Dirichlet boundary condition. Furthermore, it holds that
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