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) : {equation*} λ_α(p,q,r)(∫ₐᵇ|u|qdx)^ pq ≤{∫ₐᵇ|u'|ᵖdx+α |∫ₐᵇ|u|ʳ⁻²u\, dx|^{ p{r-1}}}, {equation*} where α , p,q,r >1 such that 2p/p+2≤ q≤ p and q/2+1≤ r ≤ q+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 αC=αC (p,q,r) such that the minimizers are even and have constant sign for α≤αC , while they are odd for α≥ αC .