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Volume 26, Issue 2
A Remark on the Existence of Positive Solution for a Class of (p,q)-Laplacian Nonlinear System with Multiple Parameters and Sign-changing Weight

S. H. Rasouli

DOI: 10.4208/jpde.v26.n2.1

J. Part. Diff. Eq., 26 (2013), pp. 99-106.

Published online: 2013-06

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  • Abstract

The paper deal with the existence of positive solution for the following (p,q)-Laplacian nonlinear system \begin{align*} \left\{  \begin{array}{ll} -Δ_pu=a(x)(α_1f(v)+β_1h(u)), & x∈Ω,\\ -Δ_qv=b(x)(α_2g(u)+β_2k(v)),& x∈Ω,\\ u=v=0,& x∈∂Ω, \end{array} \right. \end{align*} where $Δ_p$ denotes the p-Laplacian operator defined by $Δ_{p}z=div(|∇_z|^{p-2}∇z), p>1, α_1, α_2, β_1, β_2$ are positive parameters and Ω is a bounded domain in $R^N(N > 1)$ with smooth boundary ∂Ω. Here a(x) and b(x) are $C^1$ sign-changing functions that maybe negative near the boundary and f, g, h, k are C^1 nondecreasing functions such that $f, g, h, k: [0,∞)→[0,∞); f (s), g(s), h(s), k(s) > 0; s > 0$ and $lim_{n→∞}\frac{f(Mg(x)^{\frac{1}{q-1}}}{x^{p-1}}=0$ for every $M > 0$.  We discuss the existence of positive solution when $f, g, h, k, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

  • Keywords

(p q)-Laplacian nonlinear system multiple parameters sign-changing weight

  • AMS Subject Headings

35J55, 35J65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

s.h.rasouli@nit.ac.ir (S. H. Rasouli)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-26-99, author = {Rasouli , S. H.}, title = {A Remark on the Existence of Positive Solution for a Class of (p,q)-Laplacian Nonlinear System with Multiple Parameters and Sign-changing Weight}, journal = {Journal of Partial Differential Equations}, year = {2013}, volume = {26}, number = {2}, pages = {99--106}, abstract = {

The paper deal with the existence of positive solution for the following (p,q)-Laplacian nonlinear system \begin{align*} \left\{  \begin{array}{ll} -Δ_pu=a(x)(α_1f(v)+β_1h(u)), & x∈Ω,\\ -Δ_qv=b(x)(α_2g(u)+β_2k(v)),& x∈Ω,\\ u=v=0,& x∈∂Ω, \end{array} \right. \end{align*} where $Δ_p$ denotes the p-Laplacian operator defined by $Δ_{p}z=div(|∇_z|^{p-2}∇z), p>1, α_1, α_2, β_1, β_2$ are positive parameters and Ω is a bounded domain in $R^N(N > 1)$ with smooth boundary ∂Ω. Here a(x) and b(x) are $C^1$ sign-changing functions that maybe negative near the boundary and f, g, h, k are C^1 nondecreasing functions such that $f, g, h, k: [0,∞)→[0,∞); f (s), g(s), h(s), k(s) > 0; s > 0$ and $lim_{n→∞}\frac{f(Mg(x)^{\frac{1}{q-1}}}{x^{p-1}}=0$ for every $M > 0$.  We discuss the existence of positive solution when $f, g, h, k, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v26.n2.1}, url = {http://global-sci.org/intro/article_detail/jpde/5155.html} }
TY - JOUR T1 - A Remark on the Existence of Positive Solution for a Class of (p,q)-Laplacian Nonlinear System with Multiple Parameters and Sign-changing Weight AU - Rasouli , S. H. JO - Journal of Partial Differential Equations VL - 2 SP - 99 EP - 106 PY - 2013 DA - 2013/06 SN - 26 DO - http://doi.org/10.4208/jpde.v26.n2.1 UR - https://global-sci.org/intro/article_detail/jpde/5155.html KW - (p KW - q)-Laplacian nonlinear system KW - multiple parameters KW - sign-changing weight AB -

The paper deal with the existence of positive solution for the following (p,q)-Laplacian nonlinear system \begin{align*} \left\{  \begin{array}{ll} -Δ_pu=a(x)(α_1f(v)+β_1h(u)), & x∈Ω,\\ -Δ_qv=b(x)(α_2g(u)+β_2k(v)),& x∈Ω,\\ u=v=0,& x∈∂Ω, \end{array} \right. \end{align*} where $Δ_p$ denotes the p-Laplacian operator defined by $Δ_{p}z=div(|∇_z|^{p-2}∇z), p>1, α_1, α_2, β_1, β_2$ are positive parameters and Ω is a bounded domain in $R^N(N > 1)$ with smooth boundary ∂Ω. Here a(x) and b(x) are $C^1$ sign-changing functions that maybe negative near the boundary and f, g, h, k are C^1 nondecreasing functions such that $f, g, h, k: [0,∞)→[0,∞); f (s), g(s), h(s), k(s) > 0; s > 0$ and $lim_{n→∞}\frac{f(Mg(x)^{\frac{1}{q-1}}}{x^{p-1}}=0$ for every $M > 0$.  We discuss the existence of positive solution when $f, g, h, k, a(x)$ and $b(x)$ satisfy certain additional conditions. We use the method of sub-super solutions to establish our results.

S. H. Rasouli. (2019). A Remark on the Existence of Positive Solution for a Class of (p,q)-Laplacian Nonlinear System with Multiple Parameters and Sign-changing Weight. Journal of Partial Differential Equations. 26 (2). 99-106. doi:10.4208/jpde.v26.n2.1
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