### Electronic Research Archive

2020, Issue 4: 1419-1438. doi: 10.3934/era.2020075

# A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior

• Received: 01 April 2020 Revised: 01 June 2020 Published: 31 July 2020
• 35J60, 35J66, 35J92

• We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by $p$-Laplacian elliptic equations

\begin{align*} \left \{ \begin{array}{l} -\Delta_p z_1 = \lambda_1 g_1(z_2)\ \ {\rm in}\ \Omega,\\ -\Delta_p z_2 = \lambda_2 g_2(z_1)\ \ {\rm in}\ \Omega,\\ z_1 = z_2 = 0\ \ {\rm on}\ \ \partial \Omega, \end{array} \right. \end{align*}

where $\Delta_p u = \text{div}({|\nabla u|}^{p-2}\nabla u),\ 1<p<\infty$, $\lambda_1$ and $\lambda_2$ are positive parameters, $\Omega$ is the open unit ball in $\mathbb{R}^N,\ N\geq 2$.

Citation: Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior[J]. Electronic Research Archive, 2020, 28(4): 1419-1438. doi: 10.3934/era.2020075

### Related Papers:

• We prove uniqueness, existence and asymptotic behavior of positive solutions to the system coupled by $p$-Laplacian elliptic equations

\begin{align*} \left \{ \begin{array}{l} -\Delta_p z_1 = \lambda_1 g_1(z_2)\ \ {\rm in}\ \Omega,\\ -\Delta_p z_2 = \lambda_2 g_2(z_1)\ \ {\rm in}\ \Omega,\\ z_1 = z_2 = 0\ \ {\rm on}\ \ \partial \Omega, \end{array} \right. \end{align*}

where $\Delta_p u = \text{div}({|\nabla u|}^{p-2}\nabla u),\ 1<p<\infty$, $\lambda_1$ and $\lambda_2$ are positive parameters, $\Omega$ is the open unit ball in $\mathbb{R}^N,\ N\geq 2$.

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