Hardy–Hénon fractional equation with nonlinearities involving exponential critical growth

Abstract

In this paper, our goal is to study the following class of Hardy–Hénon type problems

$$\begin{aligned} \left\{ \begin{array}{rclcl}\displaystyle (-\Delta )^{1/2} u& =& \lambda |x|^{\mu } u+|x|^{\alpha }f(u)& \text{ in }& (-1,1),\\ u& =& 0& \text{ on }& \mathbb {R}\setminus (-1,1), \end{array}\right. \end{aligned}$$

when \(\mu \ge \alpha {>-1}\), and the nonlinearity f has exponential critical growth in the sense of the Trudinger-Moser inequality. This way, with an appropriate change of variable, due to the behavior of the weight \(|x|^{\alpha }\), one can obtain a version of this inequality in the radial context that allows us to treat the problem with a nonlocal framework and a critical exponent depending on \(\alpha \). When \(\alpha >0\), we have a Hénon problem and this exponent becomes larger than usual. This fact is a counterpart for Ni [37] result for the local case and \(\mathbb {R}^N\) (\(N\ge 3)\). If \(-1<\alpha <0\), we have a Hardy equation. In this case, the exponent is smaller than usual, but we treat a problem with a singularity at 0. The main difficulty is to overcome the lack of compactness inherent to problems involving nonlinearities with critical growth. For this, we apply the variational methods to control the minimax level using Moser functions (see [48]). Then, we guarantee the existence of at least one radial solution under suitable hypotheses for the constants \(\lambda , \mu ,\alpha \), as well as, on f, combined with the interaction of the spectrum of the fractional Laplacian operator via the Mountain Pass Theorem or the Linking Theorem (see [41, 42]). Thus, we study a version of problems treated in [3] for nonlinearities with exponential critical growth and in [25] in a nonlocal context.