Nonradial solutions for a conformally invariant fourth order equation in $\mathbb {R}^4$

Abstract

We consider the following Liouville equation in ${\mathbb{R}}^{4}$

$$\Delta^{2}u = 6e^{4u}\,{\rm in}\,{\mathbb{R}}^{4}, \int \limits_{{\mathbb{R}}^{4}} e^{4u}dx < \infty.$$

For each fixed $x^0 \in {\mathbb{R}}^{4}, 1 \le k \le 4, \alpha \in (1-\frac{k}{4}, 2)$ and a _j  > 0 for 1 ≤ j ≤ k, we construct a solution to the above equation with the following asymptotic behavior:

$$u(x) = - \sum_{j=1}^{k}a_j (x_j - x_j^0)^2 - \alpha\,{\rm log}\,|x|+c_0 + o(1), |x| > 1,$$

$$\int \limits_{{\mathbb{R}}^{4}} e^{4u(x)}dx= \frac{4\pi^{2}\alpha}{3}.$$