Symmetry results for a nonlocal nonlinear Poincar\'e-Wirtinger inequality

In this paper, we study the optimal constant in the nonlocal nonlinear Poincar\'e-Wirtinger inequality in $(a,b)\subset\mathbb R$: \begin{equation*} \lambda_\alpha(p,q,r){\left(\int_{a}^{b}|u|^{q}dx\right)^\frac pq}\le{\int_{a}^{b}|u'|^{p}dx+\alpha\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 $\frac q2+1\le r \le q+\frac q p$. 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 $\alpha\geq \alpha_{C}$.