Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals

Abstract

Let II be a bounded interval of R{\mathbb{R}} and λ1(I){\lambda }_{1}\left(I) denote the first eigenvalue of the nonlocal operator (−Δ)14{(-\Delta )}^{\tfrac{1}{4}} with the Dirichlet boundary. We prove that for any 0⩽α<λ1(I)0\leqslant \alpha \lt {\lambda }_{1}(I), there holds supu∈W012,2(I),‖(−Δ)14u‖22−α∥u∥22≤1∫Ieπu2dx<+∞,\mathop{\sup }\limits_{u\in {W}_{0}^{\frac{1}{2},2}(I),\Vert {\left(-\Delta )}^{\tfrac{1}{4}}u{\Vert }_{2}^{2}-\alpha {\parallel u\parallel }_{2}^{2}\le 1}\mathop{\int }\limits_{I}{e}^{\pi {u}^{2}}{\rm{d}}x\lt +\infty , and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions.

Keywords

• fractional
• extremals
• trudinger-moser inequalities
• 46e30
• 46e35