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

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 $4 < p < 6$.