Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights
It is well-known that it is a quite interesting topic to study the asymptotic expansions of entire large solutions of nonlinear elliptic equations near infinity. But very little is done. In this study, we establish the (m+1)\left(m+1)-expansions of entire kk-convex large solutions near infinity to t...
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De Gruyter
2025-03-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2025-0076 |
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| description | It is well-known that it is a quite interesting topic to study the asymptotic expansions of entire large solutions of nonlinear elliptic equations near infinity. But very little is done. In this study, we establish the (m+1)\left(m+1)-expansions of entire kk-convex large solutions near infinity to the kk-Hessian equation Sk(D2u)=b(x)f(u)inRN,{S}_{k}\left({D}^{2}u)=b\left(x)f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where m∈N+m\in {{\mathbb{N}}}_{+}, 1≤k<N⁄21\le k\lt N/2, N≥3N\ge 3, f(u)=upf\left(u)={u}^{p} (p>kp\gt k) near infinity or f(u)=up+uqf\left(u)={u}^{p}+{u}^{q} (p>kp\gt k and p>q>−1p\gt q\gt -1) near infinity. In particular, inspired by some ideas in partition theory of integer, we give a recursive formula of the coefficient of (n+1)\left(n+1)-order terms (2≤n≤m)\left(2\le n\le m) of the expansions. And if f(u)=up+uqf\left(u)={u}^{p}+{u}^{q} near infinity, we reveal the influence of the lower term of f(u)f\left(u) on the expansion of any entire large solution. |
| format | Article |
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| institution | DOAJ |
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| language | English |
| publishDate | 2025-03-01 |
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| spelling | doaj-art-1fd58014d2f94c72b024bfa881230ef62025-08-20T03:05:07ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-03-0114134536010.1515/anona-2025-0076Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weightsWan Haitao0School of Mathematics and Information Science, Shandong Technology and Business University, Yantai City, Shandong Province, P. R. ChinaIt is well-known that it is a quite interesting topic to study the asymptotic expansions of entire large solutions of nonlinear elliptic equations near infinity. But very little is done. In this study, we establish the (m+1)\left(m+1)-expansions of entire kk-convex large solutions near infinity to the kk-Hessian equation Sk(D2u)=b(x)f(u)inRN,{S}_{k}\left({D}^{2}u)=b\left(x)f\left(u)\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where m∈N+m\in {{\mathbb{N}}}_{+}, 1≤k<N⁄21\le k\lt N/2, N≥3N\ge 3, f(u)=upf\left(u)={u}^{p} (p>kp\gt k) near infinity or f(u)=up+uqf\left(u)={u}^{p}+{u}^{q} (p>kp\gt k and p>q>−1p\gt q\gt -1) near infinity. In particular, inspired by some ideas in partition theory of integer, we give a recursive formula of the coefficient of (n+1)\left(n+1)-order terms (2≤n≤m)\left(2\le n\le m) of the expansions. And if f(u)=up+uqf\left(u)={u}^{p}+{u}^{q} near infinity, we reveal the influence of the lower term of f(u)f\left(u) on the expansion of any entire large solution.https://doi.org/10.1515/anona-2025-0076(m + 1)-expansions of solutionsentire k-convex large solutionsk-hessian equations35j6035j1535b0835b40 |
| spellingShingle | Wan Haitao Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights Advances in Nonlinear Analysis (m + 1)-expansions of solutions entire k-convex large solutions k-hessian equations 35j60 35j15 35b08 35b40 |
| title | Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights |
| title_full | Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights |
| title_fullStr | Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights |
| title_full_unstemmed | Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights |
| title_short | Sharp asymptotic expansions of entire large solutions to a class of k-Hessian equations with weights |
| title_sort | sharp asymptotic expansions of entire large solutions to a class of k hessian equations with weights |
| topic | (m + 1)-expansions of solutions entire k-convex large solutions k-hessian equations 35j60 35j15 35b08 35b40 |
| url | https://doi.org/10.1515/anona-2025-0076 |
| work_keys_str_mv | AT wanhaitao sharpasymptoticexpansionsofentirelargesolutionstoaclassofkhessianequationswithweights |