Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

Francesco Della Pietra; Gianpaolo Piscitelli

doi:10.4171/rlm/883

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

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Abstract

Let us consider the following minimum problem

λ_{α} (p, r) = u \in W_{0}^{1, p} (- 1, 1) u \neq \equiv 0 min \frac{\int _{- 1}^{1} ∣ u ^{'} ∣ ^{p} d x + α \int _{- 1}^{1} ∣ u ∣ ^{r - 1} u d x ^{\frac{p}{r}}}{\int _{- 1}^{1} ∣ u ∣ ^{p} d x},

where $α \in R$ , $p \geq 2$ and $\frac{p}{2} \leq r \leq p$ . We show that there exists a critical value $α_{C} = α_{C} (p, r)$ such that the minimizers have constant sign up to $α = α_{C}$ and then they are odd when $α > α_{C}$ .

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