Jijiang Sun (Author), Lin Li (Author), Matija Cencelj (Author), Boštjan Gabrovšek (Author)

Abstract

Obravnavamo naslednji nelinearni problem Kirchhoffovega tipa ▫$$\begin{cases} - \Big (a+b \int_{\mathbb{R}^3} |\nabla u|^2 \Big) \Delta u + V(x)u = f(u), & \text{in} \quad\mathbb{R}^3 \; , \\ u \in H^1 (\mathbb{R}^3) \; , \end{cases}$$▫ kjer sta ▫$a,b > 0$▫ konstanti, nelinearni člen ▫$f$▫ je superlinearen v neskončnosti, s subkritično rastjo, ▫$V$▫ pa je zvezna in vsiljena funkcija. V primeru, ko je ▫$f$▫ liha funkcija za ▫$u$▫, dobimo z uporabo kombinacije invariantnih množic in mini-maks metode Ljusternik-Schnirelmanovega tipa neskončno mnogo rešitev s spremenljivim predznakom za ta problem. Kolikor je nam znano, je bilo doslej najdenih le malo eksistenčnih rezultatov za ta problem. Velja omeniti, da nelinearni člen ni nujno 4-superlinearen v neskončnosti, konkretno vključuje nelinearnost potenčnega tipa ▫$|u|^{p-2}u$▫ za ▫$p$▫ iz intervala ▫$(2,4]$▫.

Keywords

neskončno rešitev s spremenljivim predznakom;problemi Kirchhoffovega tipa;invariantne množice;pojemajoč tok;infinitely many sign-changing solutions;Kirchhoff type problems;invariant sets;descending flow;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UL FS - Faculty of Mechanical Engineering
UDC: 517.956
COBISS: 18506585 Link will open in a new window
ISSN: 0362-546X
Views: 400
Downloads: 240
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Other data

Secondary language: Slovenian
Secondary title: Neskončno mnogo rešitev s spremenljivim predznakom za probleme Kirchhoffovega tipa v R[sup]3
Secondary abstract: In this paper, we consider the following nonlinear Kirchhoff type problem: ▫$$\begin{cases} - \Big (a+b \int_{\mathbb{R}^3} |\nabla u|^2 \Big) \Delta u + V(x)u = f(u), & \text{in} \quad \mathbb{R}^3 \; , \\ u \in H^1 (\mathbb{R}^3) \; , \end{cases}$$▫ where ▫$a,b > 0$▫ are constants, the nonlinearity ▫$f$▫ is superlinear at infinity with subcritical growth and ▫$V$▫ is continuous and coercive. For the case when ▫$f$▫ is odd in ▫$u$▫ we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity ▫$|u|^{p-2}u$▫ with ▫$p \in (2, 4]$▫.
Secondary keywords: neskončno rešitev s spremenljivim predznakom;problemi Kirchhoffovega tipa;invariantne množice;pojemajoč tok;
Type (COBISS): Article
Pages: str. 33-54
Issue: ǂVol. ǂ186
Chronology: Sep. 2019
DOI: 10.1016/j.na.2018.10.007
ID: 11633686