The existence and concentration of ground-state solutions for a class of Kirchhoff type problems in R 3 ${\mathbb{R}^{3}}$ involving critical Sobolev exponents

Abstract We are concerned with ground-state solutions for the following Kirchhoff type equation with critical nonlinearity: { − ( ε 2 a + ε b ∫ R 3 | ∇ u | 2 ) Δ u + V ( x ) u = λ W ( x ) | u | p − 2 u + | u | 4 u in R 3 , u > 0 , u ∈ H 1 ( R 3 ) , $$\textstyle\begin{cases} - ({\varepsilon^{2}}a + \varepsilon b\int_{{\mathbb{R}^{3}}} {{{ \vert {\nabla u} \vert }^{2}}} )\Delta u + V(x)u = \lambda W(x){ \vert u \vert ^{p - 2}}u + { \vert u \vert ^{4}}u\quad {\text{in }}{\mathbb{R}^{3}} ,\\ u > 0, \quad\quad u \in{H^{1}}({\mathbb{R}^{3}}) , \end{cases} $$ where ε is a small positive parameter, a , b > 0 $a,b>0$ , λ > 0 $\lambda > 0$ , 2 0 $\varepsilon > 0$ sufficiently small, the above problem has a positive ground-state solution u ε ${u_{\varepsilon}}$ by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to Gui (Commun. Partial Differ. Equ. 21:787-820, 1996) to show that u ε ${u_{\varepsilon}}$ is concentrated around a set which is related to the set where the potential V ( x ) $V(x)$ attains its global minima or the set where the potential W ( x ) $W(x)$ attains its global maxima as ε → 0 $\varepsilon \to0$