# Standing waves for a class of Kirchhoff type problems in $${\mathbb {R}^3}$$ involving critical Sobolev exponents

• Yi He
• Gongbao Li
Article

## Abstract

We are concerned with the following Kirchhoff type equation with critical nonlinearity:
\begin{aligned} \left\{ \begin{array}{ll} - \Bigl ( {{\varepsilon ^2}a + \varepsilon b\int _{{\mathbb {R}^3}} {{{| {\nabla u} |}^2}} } \Bigr )\Delta u + V(x)u = \lambda {| u |^{p - 2}}u + {| u |^4}u{\text { in }}{\mathbb {R}^3}, \\ u > 0,u \in {H^1}({\mathbb {R}^3}), \\ \end{array} \right. \end{aligned}
where $$\varepsilon$$ is a small positive parameter, $$a,b>0$$, $$\lambda > 0$$, $$2 < p \le 4$$. Under certain assumptions on the potential V, we construct a family of positive solutions $${u_\varepsilon } \in {H^1}({\mathbb {R}^3})$$ which concentrates around a local minimum of V as $$\varepsilon \rightarrow 0$$. Although, critical growth Kirchhoff type problem
\begin{aligned} \left\{ \begin{array}{ll} - \Bigl ( {{\varepsilon ^2}a + \varepsilon b\int _{{\mathbb {R}^3}} {{{| {\nabla u} |}^2}} } \Bigr )\Delta u + V(x)u = f(u)+{u^5}{\text { in }}{\mathbb {R}^3}, \\ u > 0,u \in {H^1}({\mathbb {R}^3}) \\ \end{array} \right. \end{aligned}
has been studied in e.g. He et al. [18], where the assumption for f(u) is that $$f(u) \sim |u{|^{p - 2}}u$$ with $$4 < p < 6$$ and satisfies the Ambrosetti-Rabinowitz condition which forces the boundedness of any Palais-Smale sequence of the corresponding energy functional of the equation. As $$g(u): = \lambda {| u |^{p - 2}}u + {| u |^4u}$$ with $$2<p\le 4$$ does not satisfy the Ambrosetti-Rabinowitz condition ($$\exists \mu > 4, 0 < \mu \int _0^u {g(s)ds \le g(u)u}$$), the boundedness of Palais–Smale sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function $$g(s)/{s^3}$$ is not increasing for $$s > 0$$ prevents us from using the Nehari manifold directly as usual. Our result extends the main result in He et al. [18] concerning the existence and concentration of positive solutions to the case where $$f(u) \sim |u{|^{p - 2}}u$$ with $$4 < p < 6$$.

## Mathematics Subject Classification

Primary 35J20 35J60 35J92

## Notes

### Acknowledgments

The authors would like to express their sincere gratitude to the referee for all insightful comments and valuable suggestions, based on which the paper was revised.

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