The existence of constrained minimizers related to fractional $p$-Laplacian equations

Abstract

The existence of the solutions with prescribed $L^{p}$-norm for a fractional $p$-Laplacian equation is investigated in this paper. The obtained result is suitable for all the order of the derivative $0< s< 1$ and $p> 1$, which extends the previous results for $s=1$ or $p=2$. In particular, to the best of our knowledge, as the $L^{p}$-subcritical or $L^{p}$-critical constrained minimization problem for fractional $p$-Laplacian equation, the critical exponent $({pN+p^{2}s})/{N}$ is properly established for the first time. On one hand, using Lions Vanishing Lemma and Brézis-Lieb Lemma, the compactness of minimizing sequences for the related constrained minimization problem is derived, then based on which the existence of constrained minimizers is achieved. On the other hand, the existence of weak solution and the nonexistence result are also provided.