Concentrating Bound States for Kirchhoff type problems in ${\R^3}$ involving critical Sobolev exponents

Abstract

We study the concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth \[\left\{\begin{gathered} - \Bigl({\varepsilon ^2}a + \varepsilon b\int_{{\R^3}} {{{\left| {\nabla u} \right|}^2}} \Bigr)\Delta u + V(z)u = f(u) + {u^5}{\text{in}}{\R^3}, \hfill u \in {H^1}({\R^3}),{\text{}}u > 0{\text{in}}{\R^3}, \hfill \\ \end{gathered} \right.\] where $\varepsilon $ is a small positive parameter and $a,b > 0$ are constants, $f \in {C^1}({\R^ +},\R)$ is subcritical, $V:{\R^3} \to \R$ is a locally H\"{o}lder continuous function. We first prove that for ${\varepsilon_0} > 0$ sufficiently small, the above problem has a weak solution ${u_\varepsilon}$ with exponential decay at infinity. Moreover, ${u_\varepsilon}$ concentrates around a local minimum point of $V$ in $\Lambda $ as $\varepsilon \to 0$. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topology construct of the set where the potential $V\left(z \right)$ attains its minimum.