Existence of solutions for Kirchhoff type problems with critical nonlinearity in $${\mathbb{R}^N}$$ R N

Abstract

In this paper, we consider the existence and multiplicity of standing wave solutions of Kirchhoff type problems with critical nonlinearity in \({\mathbb{R}^N}\) : $$-\varepsilon^p \left(a + b \int\limits_{\mathbb{R}^N} \frac{1}{p}|\nabla u|^p{\rm d}x \right) \,{\rm div}(|\nabla u|^{p-2}\nabla u) + V(x)|u|^{p-2}u = K(x)|u|^{p^\ast-2}u + h(x,u),$$ for all \({(t, x) \in \mathbb{R} \times \mathbb{R}^N}\), where V(x) is a nonnegative potential, and K(x) is a bounded positive function. Under suitable assumptions, we show that this equation has at least one solution provided that \({\varepsilon < \mathcal {E}}\), for any \({m \in \mathbb{N}}\), it has m pairs of solutions if \({\varepsilon < \mathcal {E}_m}\), where \({\mathcal {E}}\) and \({\mathcal {E}_m}\) are sufficiently small positive numbers. Moreover, these solutions \({u_\varepsilon \rightarrow 0}\) in \({W^{1,p}(\mathbb{R}^N)}\) as \({\varepsilon \rightarrow 0}\).