Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN
We focus on the following elliptic system with critical Sobolev exponents: -div∇up-2∇u+m(x)up-2u=λup⁎-2u+(1/η)Gu(u,v), x∈RN; -div∇vq-2∇v+n(x)vq-2v=μvq⁎-2v+(1/η)Gv(u,v), x∈RN; u(x)>0,v(x)>0, x∈RN, where μ,λ>0,1<p≤q<N, either η∈(1,p) or η∈(q,p⁎), and critical Sobolev exponents p⁎=...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2018-01-01
|
| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2018/6458395 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832553728541982720 |
|---|---|
| author | Jing Li Caisheng Chen |
| author_facet | Jing Li Caisheng Chen |
| author_sort | Jing Li |
| collection | DOAJ |
| description | We focus on the following elliptic system with critical Sobolev exponents: -div∇up-2∇u+m(x)up-2u=λup⁎-2u+(1/η)Gu(u,v), x∈RN; -div∇vq-2∇v+n(x)vq-2v=μvq⁎-2v+(1/η)Gv(u,v), x∈RN; u(x)>0,v(x)>0, x∈RN, where μ,λ>0,1<p≤q<N, either η∈(1,p) or η∈(q,p⁎), and critical Sobolev exponents p⁎=pN/(N-p) and q⁎=qN/(N-q). Conditions on potential functions m(x),n(x) lead to no compact embedding. Relying on concentration-compactness principle, mountain pass lemma, and genus theory, the existence of solutions to the elliptic system with η∈(q,p⁎) or η∈(1,p) will be established. |
| format | Article |
| id | doaj-art-1486300432b0470aa105425d20bfa61c |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-1486300432b0470aa105425d20bfa61c2025-02-03T05:53:15ZengWileyAdvances in Mathematical Physics1687-91201687-91392018-01-01201810.1155/2018/64583956458395Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RNJing Li0Caisheng Chen1College of Science, Hohai University, Nanjing 210098, ChinaCollege of Science, Hohai University, Nanjing 210098, ChinaWe focus on the following elliptic system with critical Sobolev exponents: -div∇up-2∇u+m(x)up-2u=λup⁎-2u+(1/η)Gu(u,v), x∈RN; -div∇vq-2∇v+n(x)vq-2v=μvq⁎-2v+(1/η)Gv(u,v), x∈RN; u(x)>0,v(x)>0, x∈RN, where μ,λ>0,1<p≤q<N, either η∈(1,p) or η∈(q,p⁎), and critical Sobolev exponents p⁎=pN/(N-p) and q⁎=qN/(N-q). Conditions on potential functions m(x),n(x) lead to no compact embedding. Relying on concentration-compactness principle, mountain pass lemma, and genus theory, the existence of solutions to the elliptic system with η∈(q,p⁎) or η∈(1,p) will be established.http://dx.doi.org/10.1155/2018/6458395 |
| spellingShingle | Jing Li Caisheng Chen Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN Advances in Mathematical Physics |
| title | Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN |
| title_full | Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN |
| title_fullStr | Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN |
| title_full_unstemmed | Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN |
| title_short | Two Types of Solutions to a Class of (p,q)-Laplacian Systems with Critical Sobolev Exponents in RN |
| title_sort | two types of solutions to a class of p q laplacian systems with critical sobolev exponents in rn |
| url | http://dx.doi.org/10.1155/2018/6458395 |
| work_keys_str_mv | AT jingli twotypesofsolutionstoaclassofpqlaplaciansystemswithcriticalsobolevexponentsinrn AT caishengchen twotypesofsolutionstoaclassofpqlaplaciansystemswithcriticalsobolevexponentsinrn |