Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation

Abstract

Abstract We deal with localized semiclassical states for singularly perturbed Kirchhoff-type equation as follows: − ε 2 a + ε b ∫ R 3 ∣ ∇ v ∣ 2 d x Δ v + V ( x ) v = P ( x ) f ( v ) , x ∈ R 3 , -\left({\varepsilon }^{2}a+\varepsilon b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}| \nabla v{| }^{2}{\rm{d}}x\right)\Delta v+V\left(x)v=P\left(x)f\left(v),\hspace{1em}x\in {{\mathbb{R}}}^{3}, where V , P ∈ C 1 ( R 3 , R ) V,P\in {C}^{1}\left({{\mathbb{R}}}^{3},{\mathbb{R}}) and bounded away from zero. By applying the penalization approach together with the Nehari manifold approach in the studies of Szulkin and Weth, we obtain the existence of an infinite sequence of localized solutions of higher topological type. In addition, we also give a concrete set as the concentration position of these localized solutions. It is noted that, in our main results, f f only belongs to C ( R , R ) C\left({\mathbb{R}},{\mathbb{R}}) and does not satisfy the Ambrosetti-Rabinowitz-type condition.