Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term

Abstract In this article, we study the following Kirchhoff equation: (0.1) − stretchy="false">( a + b ‖ ∇ u L 2 ( R 3 ) stretchy="false">) mathvariant="normal">Δ V ∣ x = f width="0.1em" in width="0.33em" , -(a+b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2})\Delta u+V\left(| x| )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where xmlns:m="http://www.w3.org/1998/Math/MathML"> > 0 a,b\gt 0 , V is a positive radial potential function, and f\left(u) an asymptotically cubic term. The nonlocal term b\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u 3-homogeneous in sense that t tu{\Vert \left(tu)={t}^{3}b\Vert so it competes complicatedly with which totally different from super-cubic case. By using Miranda theorem classifying domain partitions, via gluing method variational method, prove for each integer k k equation has nodal solution U 4 {U}_{k,4}^{b} exactly 1 k+1 domains. Moreover, show energy of strictly increasing any sequence { n } → \left\{{b}_{n}\right\}\to {0}_{+}, up to subsequence, {U}_{k,4}^{{b}_{n}} converges strongly {U}_{k,4}^{0} H {H}^{1}\left({{\mathbb{R}}}^{3}) also domains solves classical Schrödinger − . -a\Delta )u=f\left(u)\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}. Our results extend ones Deng et al. case