Topological degree for solutions of fourth order mean field equations

Abstract

We consider the following fourth order mean field equation with Navier boundary conditionwhere h is a C 2,β positive function, is a bounded and smooth domain in R 4 . We prove that for ρ ∈ (32mσ 3 , 32(m + 1)σ 3 ) the degree-counting formula for (*) is given bywhere χ( ) is the Euler characteristic of . Similar result is also proved for the corresponding Dirichlet problem 2 u = ρ h(x)e u he u in , u = ∇u = 0 on ∂ . (**)

Existence and multiplicity results for a fourth order mean field equation

“…Under the stronger condition that χ(Ω) 0, where χ(Ω) denotes the Euler characteristic of Ω, the above theorem has been proved in [22]. Their proof which is drastically different from ours, uses a topological degree argument.…”

Singular limits for the bi-Laplacian operator with exponential nonlinearity in $\mathbb R^{4}$

“…Lin and J.-C. Wei in [26], where they show that, when blow-up occurs for (1.1) as ρ → 0, then it is located at a finite number of peaks, each peak being isolated and carrying the energy 64π 2 (at a peak, u → +∞ and outside a peak, u is bounded). See [27] and [28] for related results.…”

Morse theory for a fourth order elliptic equation with exponential nonlinearity

“…q.e.d. In the next Lemma we show a criterion which implies the situation described in the first condition in (13).…”

“…q.e.d. In the next Lemma we show a criterion which implies the situation described in the first condition in (13). Lemma 1.11 ([8, Lemma 2.3]) Let l be a given positive integer, and suppose that ε and r are positive numbers.…”