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

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. , 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.  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

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