Symmetry results for a nonlocal nonlinear Poincaré–Wirtinger inequality

In this paper, we study the optimal constant in the nonlocal nonlinear Poincaré–Wirtinger inequality in (a,b)\subset\mathbb{R} : \begin{equation*}\lambda_\alpha(p,q,r)\bigg(\int_{a}^{b}|u|^{q}dx\bigg)^\fracpq \le{\int_{a}^{b}|u'|^{p}dx+\alpha\bigg|\int_{a}^{b}|u|^{r-2}u\, dx\bigg|^{\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\geq \alpha_{C} .