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Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
- Source :
- Advances in Nonlinear Analysis, Vol 12, Iss 1, Pp 305-330 (2023)
- Publication Year :
- 2023
- Publisher :
- De Gruyter, 2023.
-
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.
Details
- Language :
- English
- ISSN :
- 2191950X
- Volume :
- 12
- Issue :
- 1
- Database :
- Directory of Open Access Journals
- Journal :
- Advances in Nonlinear Analysis
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.2b44dc2b48034fb2a82a2035db642a0e
- Document Type :
- article
- Full Text :
- https://doi.org/10.1515/anona-2022-0323