Volume 25, Issue 1
A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn

Liang Zhao

J. Part. Diff. Eq., 25 (2012), pp. 90-102.

Published online: 2012-03

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  • Abstract

We establish sufficient conditions under which the quasilinear equation  $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.

  • Keywords

Moser-Trudinger inequality exponential growth

  • AMS Subject Headings

58J05, 58E30, 35J60, 35B33, 35J20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

liangzhao@bnu.edu.cn (Liang Zhao)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-25-90, author = {Zhao , Liang}, title = {A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn}, journal = {Journal of Partial Differential Equations}, year = {2012}, volume = {25}, number = {1}, pages = {90--102}, abstract = {

We establish sufficient conditions under which the quasilinear equation  $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v25.n1.7}, url = {http://global-sci.org/intro/article_detail/jpde/5177.html} }
TY - JOUR T1 - A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn AU - Zhao , Liang JO - Journal of Partial Differential Equations VL - 1 SP - 90 EP - 102 PY - 2012 DA - 2012/03 SN - 25 DO - http://doi.org/10.4208/jpde.v25.n1.7 UR - https://global-sci.org/intro/article_detail/jpde/5177.html KW - Moser-Trudinger inequality KW - exponential growth AB -

We establish sufficient conditions under which the quasilinear equation  $$-div(|∇u|^{n-2}∇u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^β}+εh(x) in \mathbb{R}^n,$$ has at least two nontrivial weak solutions in $W^{1,n} (\mathbb{R}^n)$ when ε > 0 is small enough, 0≤β < n, V is a continuous potential, f(x,u) behaves like $exp{γ|u|^{n/(n-1)}}$ as $|u|→∞$ for some γ > 0 and h≢ 0 belongs to the dual space of $W^{1,n} (\mathbb{R}^n)$.

Liang Zhao. (2019). A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in Rn. Journal of Partial Differential Equations. 25 (1). 90-102. doi:10.4208/jpde.v25.n1.7
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