Existence and multiplicity of solutions for Kirchhoff-type equation with radial potentials in $${\mathbb{R}^{3}}$$ R 3

Abstract

In this paper, we deal with the Kirchhoff-type equation $$\left\{\begin{array}{l@{\quad}l} \displaystyle-\left[1+\int\limits_{{\mathbb{R}}^3} \left(\left|\nabla u\right|^2+V(x)u^2\right){\rm d}x \right]\left[\Delta u+V(x)u\right] = \lambda Q(x) f(u), \quad x\in \ {\mathbb{R}}^3, \quad \quad \quad {(\rm P)_\lambda}\\ u(x)\rightarrow 0, \quad {\rm as} \ |x|\rightarrow \infty , \end{array}\right.$$ where $${\lambda > 0}$$ , V and Q are radial functions, which can be vanishing or coercive at infinity. With assumptions on f just in a neighborhood of the origin, existence and multiplicity of nontrivial radial solutions are obtained via variational methods. In particular, if f is sublinear and odd near the origin, we obtain infinitely many solutions of $${(\rm P)_\lambda}$$ for any $${\lambda > 0}$$ .