The Existence of Multiple Solutions for Nonhomogeneous Kirchhoff Type Equations in ℝ 3

Abstract

We are concerned with the existence of multiple solutions to the nonhomogeneous Kirchhoff type equation $-(a+b{\int }_{{ℝ}^{\mathrm{3}}}\mathrm{‍}|\nabla u{|}^{\mathrm{2}})\mathrm{\Delta }u+u=|u{|}^{p-\mathrm{1}}u+h(x)\mathrm{}\mathrm{}\text{in}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{ℝ}^{\mathrm{3}},$ where $a,b$ are positive constants, $p\in (\mathrm{1,5}),\mathrm{0}⩽h(x)=h(|x|)\in C^{\mathrm{1}}({ℝ}^{\mathrm{3}})\cap L^{\mathrm{2}}({ℝ}^{\mathrm{3}})$, we can find a constant $m_p>\mathrm{0}$ such that for all $p\in (\mathrm{1,5})$ the equation has at least two radial solutions provided .