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)−(a+b‖∇u‖L2(R3)2)Δu+V(∣x∣)u=f(u)inR3,-(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 a,b>0a,b\gt 0, VV is a positive radial potential function, and f(u)f\left(u) is an asymptotically cubic term. The nonlocal term b‖∇u‖L2(R3)2Δub\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u is 3-homogeneous in the sense that b‖∇tu‖L2(R3)2Δ(tu)=t3b‖∇u‖L2(R3)2Δub\Vert \nabla tu{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta \left(tu)={t}^{3}b\Vert \nabla u{\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{3})}^{2}\Delta u, so it competes complicatedly with the asymptotically cubic term f(u)f\left(u), which is totally different from the super-cubic case. By using the Miranda theorem and classifying the domain partitions, via the gluing method and variational method, we prove that for each positive integer kk, equation (0.1) has a radial nodal solution Uk,4b{U}_{k,4}^{b}, which has exactly k+1k+1 nodal domains. Moreover, we show that the energy of Uk,4b{U}_{k,4}^{b} is strictly increasing in kk, and for any sequence {bn}→0+,\left\{{b}_{n}\right\}\to {0}_{+}, up to a subsequence, Uk,4bn{U}_{k,4}^{{b}_{n}} converges strongly to Uk,40{U}_{k,4}^{0} in H1(R3){H}^{1}\left({{\mathbb{R}}}^{3}), where Uk,40{U}_{k,4}^{0} also has k+1k+1 nodal domains exactly and solves the classical Schrödinger equation: −aΔu+V(∣x∣)u=f(u)inR3.-a\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}. Our results extend the ones in Deng et al. from the super-cubic case to the asymptotically cubic case.

Keywords

• kirchhoff equation
• nodal solutions
• miranda theorem
• gluing method
• 35j20
• 35j60
• 35q40