JournalsrlmVol. 31, No. 1pp. 131–150

Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

  • Francesco Della Pietra

    Università degli Studi di Napoli Federico II, Italy
  • Gianpaolo Piscitelli

    Università degli Studi di Cassino e del Lazio Meridionale, Italy
Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems cover
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Abstract

Let us consider the following minimum problem

λα(p,r)=minuW01,p(1,1)u≢011updx+α11ur1udxpr11updx,\lambda_\alpha(p,r)=\min_{\substack{u\in W_{0}^{1,p}(-1,1)\\u\not\equiv0}}\frac{\int_{-1}^{1}|u'|^{p}dx+\alpha\left|\int_{-1}^{1}|u|^{r-1}u\, dx\right|^{\frac pr}}{\int_{-1}^{1}|u|^{p}dx},

where αR\alpha\in\mathbb R, p2p\ge 2 and p2rp\frac p2 \le r \le p. We show that there exists a critical value αC=αC(p,r)\alpha_C=\alpha_C (p,r) such that the minimizers have constant sign up to α=αC\alpha=\alpha_{C} and then they are odd when α>αC\alpha > \alpha_{C}.

Cite this article

Francesco Della Pietra, Gianpaolo Piscitelli, Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 1, pp. 131–150

DOI 10.4171/RLM/883