Theory of “Critical Points at Infinity” and a Resonant Singular Liouville-Type Equation

Abstract

Abstract We consider the following Liouville-type equation on domains of ℝ 2 ${\mathbb{R}^{2}}$ under Dirichlet boundary conditions: { - Δ ⁢ u = ϱ ⁢ K ⁢ e u ∫ Ω K ⁢ e u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\varrho\frac{Ke^{u% }}{\int_{\Omega}Ke^{u}}&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.$ where ϱ ∈ ℝ ${\varrho\in\mathbb{R}}$ and K is a smooth nonnegative function having N zeros q 1 , … , q N ${q_{1},\ldots,q_{N}}$ , which takes in a neighborhood of a zero q j ${q_{j}}$ the following form: K ⁢ ( x ) = K j ⁢ ( x ) ⁢ | x - q j | 2 ⁢ γ j   with ⁢ K j ⁢ ( x ) > 0 ⁢ and ⁢ γ i ∈ ℝ ⁢ such that ⁢ 0 < γ j := γ j ⁢ ( q j ) ∉ ℕ . $K(x)=K_{j}(x)\lvert x-q_{j}\rvert^{2\gamma_{j}}\quad\text{with }K_{j}(x)>0% \text{ and }\gamma_{i}\in\mathbb{R}\text{ such that }0<\gamma_{j}:=\gamma_{j}(% q_{j})\notin\mathbb{N}.$ Using some dynamical and topological tools from the “critical point theory at infinity” of Bahri, we study the critical points at infinity of the related variational problem. Then we derive from our analysis some existence results in the so-called resonant case, that is, when the parameter ϱ is of the form ∑ i = 1 σ 8 ⁢ π ⁢ ( 1 + γ i ) + ∑ i = σ + 1 m 8 ⁢ π ${\sum_{i=1}^{\sigma}8\pi(1+\gamma_{i})+\sum_{i=\sigma+1}^{m}8\pi}$ for a subset ( q i 1 , … , q i σ ) ${(q_{i_{1}},\ldots,q_{i_{\sigma}})}$ of Σ := { q 1 , … , q N } ${\Sigma:=\{q_{1},\ldots,q_{N}\}}$ . In particular, we provide an Euler–Poincaré-type criterium for existence of solutions.