Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity

TLDR: In this article, the existence of nonnegative solutions with negative energy was established by using Ekeland's variational principle, where the main feature consists in the presence of a (possibly degenerate) Kirchhoff model, combined with a critical Trudinger-Moser nonlinearity.

Abstract: This paper is concerned with the existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity: $$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy\right) (-\Delta )^{s}_{N/s}u=f(x,u)\,\, \ &{}\quad \mathrm{in}\ \Omega ,\\ u=0\ \ \ \ &{}\quad \mathrm{in}\ {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. } \end{aligned}$$ where $$(-\Delta )^{s}_{N/s}$$ is the fractional N / s-Laplacian operator, $$N\ge 1$$ , $$s\in (0,1)$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary, $$M:{\mathbb {R}}^+_0\rightarrow {\mathbb {R}}^+_0$$ is a continuous function, and $$f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}} $$ is a continuous function behaving like $$\exp (\alpha t^{2})$$ as $$t\rightarrow \infty $$ for some $$\alpha >0$$ . We first obtain the existence of a ground state solution with positive energy by using minimax techniques combined with the fractional Trudinger–Moser inequality. Next, the existence of nonnegative solutions with negative energy is established by using Ekeland’s variational principle. The main feature of this paper consists in the presence of a (possibly degenerate) Kirchhoff model, combined with a critical Trudinger–Moser nonlinearity.