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)⊂ℝ: λα(p,q,r)(∫ba|u|qdx)pq≤∫ba|u′|pdx+α∣∣∣∣∫ba|u|r−2udx∣∣∣∣pr−1, where α∈ℝ, p,q,r>1 such that 2pp+2≤q≤p and q2+1≤r≤q+qp. 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.