k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
In this article, our main aim is to investigate the existence of radial kk-convex solutions for the following Dirichlet system with kk-Hessian operators: Sk(D2u)=λ1ν1(∣x∣)(−u)p1(−v)q1inℬ(R),Sk(D2v)=λ2ν2(∣x∣)(−u)p2(−v)q2inℬ(R),u=v=0on∂ℬ(R).\left\{\begin{array}{ll}{S}_{k}\left({D}^{2}u)={\lambda }_{1}...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2024-03-01
|
| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2023-0136 |
| _version_ | 1797258416283975680 |
|---|---|
| author | He Xingyue Gao Chenghua Wang Jingjing |
| author_facet | He Xingyue Gao Chenghua Wang Jingjing |
| author_sort | He Xingyue |
| collection | DOAJ |
| description | In this article, our main aim is to investigate the existence of radial kk-convex solutions for the following Dirichlet system with kk-Hessian operators: Sk(D2u)=λ1ν1(∣x∣)(−u)p1(−v)q1inℬ(R),Sk(D2v)=λ2ν2(∣x∣)(−u)p2(−v)q2inℬ(R),u=v=0on∂ℬ(R).\left\{\begin{array}{ll}{S}_{k}\left({D}^{2}u)={\lambda }_{1}{\nu }_{1}\left(| x| ){\left(-u)}^{{p}_{1}}{\left(-v)}^{{q}_{1}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ {S}_{k}\left({D}^{2}v)={\lambda }_{2}{\nu }_{2}\left(| x| ){\left(-u)}^{{p}_{2}}{\left(-v)}^{{q}_{2}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ u=v=0& {\rm{on}}\hspace{1em}\partial {\mathcal{ {\mathcal B} }}\left(R).\end{array}\right. Here, up1vq1{u}^{{p}_{1}}{v}^{{q}_{1}} is called a Lane-Emden type nonlinearity. The weight functions ν1,ν2∈C([0,R],[0,∞)){\nu }_{1},{\nu }_{2}\in C\left(\left[0,R],\left[0,\infty )) with ν1(r)>0<ν2(r){\nu }_{1}\left(r)\gt 0\lt {\nu }_{2}\left(r) for all r∈(0,R]r\in \left(0,R], p1,q2{p}_{1},{q}_{2} are nonnegative and q1,p2{q}_{1},{p}_{2} are positive exponents, ℬ(R)={x∈RN:∣x∣<R}{\mathcal{ {\mathcal B} }}\left(R)=\left\{x\in {{\mathbb{R}}}^{N}:| x| \lt R\right\}, N≥2N\ge 2 is an integer, N2≤k≤N\frac{N}{2}\le k\le N. In order to achieve our main goal, we first study the existence of radial kk-convex solutions of the above-mentioned systems with general nonlinear terms by using the upper and lower solution method and Leray-Schauder degree. Based on this, by constructing a continuous curve, which divides the first quadrant into two disjoint sets, we obtain the existence and multiplicity of radial kk-convex solutions for the system depending on the parameters λ1{\lambda }_{1}, λ2{\lambda }_{2} and the continuous curve. |
| first_indexed | 2024-04-24T22:53:11Z |
| format | Article |
| id | doaj.art-fd21edd51b22465bac510336a17b2a3a |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-04-24T22:53:11Z |
| publishDate | 2024-03-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-fd21edd51b22465bac510336a17b2a3a2024-03-18T10:27:16ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2024-03-0113136238710.1515/anona-2023-0136k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearitiesHe Xingyue0Gao Chenghua1Wang Jingjing2School of Mathematics and Statistics, Northwest Normal University, 730070, Lanzhou, P. R. ChinaSchool of Mathematics and Statistics, Lanzhou City University, 730070, Lanzhou, P. R. ChinaSchool of Information Engineering, Lanzhou City University, 730070, Lanzhou, P. R. ChinaIn this article, our main aim is to investigate the existence of radial kk-convex solutions for the following Dirichlet system with kk-Hessian operators: Sk(D2u)=λ1ν1(∣x∣)(−u)p1(−v)q1inℬ(R),Sk(D2v)=λ2ν2(∣x∣)(−u)p2(−v)q2inℬ(R),u=v=0on∂ℬ(R).\left\{\begin{array}{ll}{S}_{k}\left({D}^{2}u)={\lambda }_{1}{\nu }_{1}\left(| x| ){\left(-u)}^{{p}_{1}}{\left(-v)}^{{q}_{1}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ {S}_{k}\left({D}^{2}v)={\lambda }_{2}{\nu }_{2}\left(| x| ){\left(-u)}^{{p}_{2}}{\left(-v)}^{{q}_{2}}& {\rm{in}}\hspace{1em}{\mathcal{ {\mathcal B} }}\left(R),\\ u=v=0& {\rm{on}}\hspace{1em}\partial {\mathcal{ {\mathcal B} }}\left(R).\end{array}\right. Here, up1vq1{u}^{{p}_{1}}{v}^{{q}_{1}} is called a Lane-Emden type nonlinearity. The weight functions ν1,ν2∈C([0,R],[0,∞)){\nu }_{1},{\nu }_{2}\in C\left(\left[0,R],\left[0,\infty )) with ν1(r)>0<ν2(r){\nu }_{1}\left(r)\gt 0\lt {\nu }_{2}\left(r) for all r∈(0,R]r\in \left(0,R], p1,q2{p}_{1},{q}_{2} are nonnegative and q1,p2{q}_{1},{p}_{2} are positive exponents, ℬ(R)={x∈RN:∣x∣<R}{\mathcal{ {\mathcal B} }}\left(R)=\left\{x\in {{\mathbb{R}}}^{N}:| x| \lt R\right\}, N≥2N\ge 2 is an integer, N2≤k≤N\frac{N}{2}\le k\le N. In order to achieve our main goal, we first study the existence of radial kk-convex solutions of the above-mentioned systems with general nonlinear terms by using the upper and lower solution method and Leray-Schauder degree. Based on this, by constructing a continuous curve, which divides the first quadrant into two disjoint sets, we obtain the existence and multiplicity of radial kk-convex solutions for the system depending on the parameters λ1{\lambda }_{1}, λ2{\lambda }_{2} and the continuous curve.https://doi.org/10.1515/anona-2023-0136k-hessian operatorlane-emden nonlinearityk-convex solutionsexistenceleray-schauder degreelower and upper solution method34b1534b1835j4735j96 |
| spellingShingle | He Xingyue Gao Chenghua Wang Jingjing k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities Advances in Nonlinear Analysis k-hessian operator lane-emden nonlinearity k-convex solutions existence leray-schauder degree lower and upper solution method 34b15 34b18 35j47 35j96 |
| title | k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities |
| title_full | k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities |
| title_fullStr | k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities |
| title_full_unstemmed | k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities |
| title_short | k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities |
| title_sort | k convex solutions for multiparameter dirichlet systems with k hessian operator and lane emden type nonlinearities |
| topic | k-hessian operator lane-emden nonlinearity k-convex solutions existence leray-schauder degree lower and upper solution method 34b15 34b18 35j47 35j96 |
| url | https://doi.org/10.1515/anona-2023-0136 |
| work_keys_str_mv | AT hexingyue kconvexsolutionsformultiparameterdirichletsystemswithkhessianoperatorandlaneemdentypenonlinearities AT gaochenghua kconvexsolutionsformultiparameterdirichletsystemswithkhessianoperatorandlaneemdentypenonlinearities AT wangjingjing kconvexsolutionsformultiparameterdirichletsystemswithkhessianoperatorandlaneemdentypenonlinearities |