Existence of multiple positive solutions for singular p-q-Laplacian
problems with critical nonlinearities
Abstract
In this article, we consider the following p-q-Laplacian system with
singular and critical nonlinearity \begin{equation*}
\left \{
\begin{array}{lllll}
-\Delta_{p}u-\Delta_{q}u=\frac{h_{1}(x)}{u^{r}}+\lambda\frac{\alpha}{\alpha+\beta}u^{\alpha-1}v^{\beta}
\ \ in\
\Omega ,\\
-\Delta_{p}v-\Delta_{q}v=\frac{h_{2}(x)}{v^{r}}+\lambda\frac{\beta}{\alpha+\beta}u^{\alpha}v^{\beta-1}
\ \ in\
\Omega, \\
u,v>0 \ \ \
\ \ \ in \
\Omega, \ \
\ \ \ u=v=0
\ \ \ \
\ \ \ on \
\partial\Omega,
\end{array} \right.
\end{equation*} where Ω is a bounded domain in
$\mathbb {R}^{n}$ with smooth boundary
$\partial\Omega$.
$11,\lambda\in(0,\Lambda_{*})$
is parameter with $\Lambda _{*}$ is a positive
constant and $h_{1}(x),h_{2}(x)\in
L^{\infty},h_{1}(x),h_{2}(x)>0$.
We show the existence and multiplicity of weak solution of equation
above for suitable range of $\lambda$.
, 
