Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^N
We prove a Liouville-type theorem for stable solution of the singular quasilinear elliptic equations $$\displaylines{ -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)|u|^{q-1}u, \quad \text{in } \mathbb{R}^N, \cr -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)e^u, \quad \text{in } \math...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2018-03-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2018/81/abstr.html |
| _version_ | 1818480982338043904 |
|---|---|
| author | Caisheng Chen Hongxue Song Hongwei Yang |
| author_facet | Caisheng Chen Hongxue Song Hongwei Yang |
| author_sort | Caisheng Chen |
| collection | DOAJ |
| description | We prove a Liouville-type theorem for stable solution of the singular
quasilinear elliptic equations
$$\displaylines{
-\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)|u|^{q-1}u, \quad
\text{in } \mathbb{R}^N, \cr
-\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)e^u, \quad
\text{in } \mathbb{R}^N
}$$
where $2\le p<N, -\infty<a<(N-p)/p$ and the function f(x) is continuous
and nonnegative in $\mathbb{R}^N\setminus\{0\}$ such that
$f(x)\ge c_0|x|^{b}$ as $|x|\ge R_0$, with $b>-p(1+a)$ and $c_0>0$.
The results hold for $1\le p-1<q=q_c(p,N,a,b)$ in the first equation,
and for $2\le N<q_0(p,a,b)$ in the second equation.
Here $q_0$ and $q_c$ are exponents, which are always larger than the
classical critical ones and depend on the parameters a,b. |
| first_indexed | 2024-12-10T11:29:18Z |
| format | Article |
| id | doaj.art-f7160ddf35be4424bca71e522534819d |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-10T11:29:18Z |
| publishDate | 2018-03-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-f7160ddf35be4424bca71e522534819d2022-12-22T01:50:39ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912018-03-01201881,111Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^NCaisheng Chen0Hongxue Song1Hongwei Yang2 Hohai Univ., Nanjing, China Hohai Univ., Nanjing, China Shandong Univ. of Science and Tech., Qingdao, China We prove a Liouville-type theorem for stable solution of the singular quasilinear elliptic equations $$\displaylines{ -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)|u|^{q-1}u, \quad \text{in } \mathbb{R}^N, \cr -\text{div}(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=f(x)e^u, \quad \text{in } \mathbb{R}^N }$$ where $2\le p<N, -\infty<a<(N-p)/p$ and the function f(x) is continuous and nonnegative in $\mathbb{R}^N\setminus\{0\}$ such that $f(x)\ge c_0|x|^{b}$ as $|x|\ge R_0$, with $b>-p(1+a)$ and $c_0>0$. The results hold for $1\le p-1<q=q_c(p,N,a,b)$ in the first equation, and for $2\le N<q_0(p,a,b)$ in the second equation. Here $q_0$ and $q_c$ are exponents, which are always larger than the classical critical ones and depend on the parameters a,b.http://ejde.math.txstate.edu/Volumes/2018/81/abstr.htmlSingular quasilinear elliptic equationstable solutionscritical exponentsLiouville type theorems |
| spellingShingle | Caisheng Chen Hongxue Song Hongwei Yang Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^N Electronic Journal of Differential Equations Singular quasilinear elliptic equation stable solutions critical exponents Liouville type theorems |
| title | Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^N |
| title_full | Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^N |
| title_fullStr | Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^N |
| title_full_unstemmed | Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^N |
| title_short | Liouville-type theorems for stable solutions of singular quasilinear elliptic equations in R^N |
| title_sort | liouville type theorems for stable solutions of singular quasilinear elliptic equations in r n |
| topic | Singular quasilinear elliptic equation stable solutions critical exponents Liouville type theorems |
| url | http://ejde.math.txstate.edu/Volumes/2018/81/abstr.html |
| work_keys_str_mv | AT caishengchen liouvilletypetheoremsforstablesolutionsofsingularquasilinearellipticequationsinrn AT hongxuesong liouvilletypetheoremsforstablesolutionsofsingularquasilinearellipticequationsinrn AT hongweiyang liouvilletypetheoremsforstablesolutionsofsingularquasilinearellipticequationsinrn |