Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

Abstract

Let us consider the following minimum problem \[ λ_α(p,r)= \min_{\substack{u\in W_{0}^{1,p}(-1,1)\\ u\not\equiv0}}\dfrac{\displaystyle\int_{-1}^{1}|u'|^{p}dx+α\left|\int_{-1}^{1}|u|^{r-1}u\, dx\right|^{\frac pr}}{\displaystyle\int_{-1}^{1}|u|^{p}dx}, \] where $\alpha \in \mathbb{R}$, $p\ge 2$ and $\frac{p}{2}\le r\le p$. We show that there exists a critical value ${\alpha }_C={\alpha }_C(p,r)$ such that the minimizers have constant sign up to $\alpha ={\alpha }_C$ and then they are odd when $\alpha >{\alpha }_C$.

arXiv commentarXiv admin note: text overlap with arXiv:1603.06517

References in corpus (1)

• A nonlocal anisotropic eigenvalue problem

Cited by in corpus (1)

• The Pseudo-orthogonality for Graph $1$-Laplacian Eigenvectors and Applications to Higher Cheeger Constants and Data Clustering