Symmetry results for a nonlocal nonlinear Poincaré–Wirtinger inequality

Abstract

In this paper, we study the optimal constant in the nonlocal nonlinear Poincaré–Wirtinger inequality in $(a,b)\subset \mathbb{R}$:

where $\alpha \in \mathbb{R}$, $p,q,r>1$ such that $\frac{2p}{p+2}\le q\le p$ and $\frac{q}{2}+1\le r\le q+\frac{q}{p}$. This problem admits a variational characterization in the nonlocal setting, as the associated Euler–Lagrange equation involves an integral term depending on the unknown function over the entire interval of definition.
We prove the existence of a critical value ${\alpha }_C={\alpha }_C(p,q,r)$ such that the minimizers are even and have constant sign for $\alpha \le {\alpha }_C$, while they are odd for $\alpha \ge {\alpha }_C$.