Multiplicity of concentrating positive solutions for nonlinear Kirchhoff type problems with critical growth

ABSTRACT

In this paper, we study the following semilinear Kirchhoff type equation: $-\left({ϵ}^{2}a+ϵb{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\right)\mathrm{\Delta }u+V\left(x\right)u=f\left(u\right)+u^5,u>0,u\in H^1\left({\mathbb{R}}^3\right),$ where $ϵ>0$ is a parameter, a, b are positive constants, V and f are continuous functions. Under certain assumptions on V and f, we investigate the relation between the number of solutions and the topology of the set where V attains its local minimum for small ε. We also describe the concentration phenomena of solutions as $ϵ\to 0$. The proof is based on the method of Nehari manifold, penalization techniques and Ljusternik–Schnirelmann category theory.

Keywords:

• Kirchhoff type equation
• critical growth
• Ljusternik–Schnirelmann category
• penalization method

2010 Mathematics Subject Classifications:

• 32J20
• 35J60
• 35J92

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 11601204, 11671077, 11571140), Jiangsu Overseas Visiting Scholar Program for University Prominent Young and Middle-aged Teachers and President, Qinglan Project, Fellowship of Outstanding Young Scholars of Jiangsu Province (BK20160063), the Six big talent peaks project in Jiangsu Province (XYDXX-015) and Natural Science Foundation of Jiangsu Province (BK20150478).