On a class of Kirchhoff problems via local mountain pass

Ambrosio, Vincenzo (author), Repovš, Dušan (author)

In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems: $$\begin{cases}-(\varepsilon^2a+\varepsilon b\int _{\mathbb{R}^3}|\nabla u|^2 dx) \Delta u + V(x)u = f(u)+\gamma u^5 & \text{in} \; \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), \quad u>0 & \text{in} \; \mathbb{R}^3, \end{cases}$$ where $\varepsilon>0$ is a small parameter, $a,b>0$ are constants, $\gamma \in {0,1}$, $V$ is a continuous positive potential with a local minimum, and $f$ is a superlinear continuous function with subcritical growth. The main results are obtained through suitable variational and topological arguments. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. Our theorems extend and improve in several directions the studies made in (Adv. Nonlinear Stud. 14 (2014), 483-510; J. Differ. Equ. 252 (2012), 1813-1834; J. Differ. Equ. 253 (2012), 2314-2351).

• Language: English
• Keywords: Kirchhoff problems, penalization method, Ljusternik-Schnirelmann theory, critical growth, supercritical exponent
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2022
• Number of pages: Str. 1-43
• Numbering: Vol. 126, iss.1-2
• PID: 20.500.12556/RUL-133972
• UDC: 517.951
• ISSN on article: 0921-7134
• DOI: 10.3233/ASY-201660
• COBISS.SI-ID: 43614723
• Publication date in RUL: 21.12.2021

AMBROSIO, Vincenzo and REPOVŠ, Dušan, 2022, On a class of Kirchhoff problems via local mountain pass. Asymptotic analysis [online]. 2022. Vol. 126, no. 1–2, p. 1–43. [Accessed 15 April 2025]. DOI 10.3233/ASY-201660. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=133972

Record is a part of a journal

• Title: Asymptotic analysis
• Shortened title: Asymptot. anal.
• Publisher: IOS Press
• ISSN: 0921-7134
• COBISS.SI-ID: 25030144