Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent

AbstractIn this paper, we consider the following semilinear Kirchhoff type equation $$\left\{\begin{array}{ll}-\left(\epsilon^2a + \epsilon b \int \limits_{\mathbb{R}^3}|\nabla u|^2 \right) \triangle {u}+V(x)u = \mu K(x)|u|^{p-1}u + Q(x)|u|^4u, \,\, \mathrm{in}\, \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), \,\, u > 0,\end{array}\right.$$-ϵ2a+ϵb∫R3|∇u|2▵u+V(x)u=μK(x)|u|p-1u+Q(x)|u|4u,inR3,u∈H1(R3),u>0,where $${\epsilon > 0}$$ϵ>0 is a small parameter, $${p \in [3,5)}$$p∈[3,5), a, b are positive constants, μ > 0 is a parameter, and the nonlinear growth of |u|4u reaches the Sobolev critical exponent since 2* = 6 for three spatial dimensions. We prove the existence of a positive ground state solution $${u_\epsilon}$$uϵ with exponential decay at infinity for μ > 0 and $${\epsilon}$$ϵ sufficiently small under some suitable conditions on the nonnegative functions V, K and Q. Moreover, $${u_\epsilon}$$uϵ concentrates around a global minimum point of V as $${\epsilon \rightarrow 0^+}$$ϵ→0+. The methods used here are based on the concentration-compactness principle of Lions.