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 $(a,b)\subset \mathbb{R}$: \begin{equation*} λ_α(p,q,r){\left(\int_{a}^{b}|u|^{q}dx\right)^\frac pq}\le{\int_{a}^{b}|u'|^{p}dx+α\left|\int_{a}^{b}|u|^{r-2}u\, dx\right|^{\frac p{r-1}}}, \end{equation*}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 $\alpha \ge {\alpha }_C$.