Advances in Nonlinear Analysis (Mar 2022)
Multiple nodal solutions of the Kirchhoff-type problem with a cubic term
Abstract
In this article, we are interested in the following Kirchhoff-type problem (0.1)−a+b∫RN∣∇u∣2dxΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN),\left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where a,b>0,N=2a,b\gt 0,N=2 or 3, the potential function VV is radial and bounded from below by a positive number. Because the nonlocal b∣∇u∣L2(RN)2Δub| \nabla u\hspace{-0.25em}{| }_{{L}^{2}\left({{\mathbb{R}}}^{N})}^{2}\Delta u is 3-homogeneous which is in complicated competition with the nonlinear term ∣u∣2u| u\hspace{-0.25em}{| }^{2}u. This causes that not all function in H1(RN){H}^{1}\left({{\mathbb{R}}}^{N}) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer kk, equation (0.1) admits a radial nodal solution Uk,4b{U}_{k,4}^{b} having exactly kk nodes. Moreover, we show that the energy of Uk,4b{U}_{k,4}^{b} is strictly increasing in kk and for any sequence {bn}\left\{{b}_{n}\right\} with bn→0+,{b}_{n}\to {0}_{+}, up to a subsequence, Uk,4bn{U}_{k,4}^{{b}_{n}} converges to Uk,40{U}_{k,4}^{0} in H1(RN){H}^{1}\left({{\mathbb{R}}}^{N}), which is a radial nodal solution with exactly kk nodes of the classical Schrödinger equation −aΔu+V(∣x∣)u=∣u∣2uinRN,u∈H1(RN).\left\{\begin{array}{l}-a\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}).\end{array}\right. Our results extend the existence result from the super-cubic case to the cubic case.
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