Abstract:
The problem
$$
\begin{cases}
\Delta_{m,g}u+b(x)u^{m^*-1}+f(x,u)=0\quad\text{in}\ \Omega,
\\
u\geqslant0\quad\text{in}\ \Omega,
\\
u=0\quad\text{on}\ \partial\Omega,
\end{cases}
$$
is investigated, where
$$
\Delta_{m,g}u=\nabla_i(g(x)|\nabla u|^{m-2}\nabla_iu),
$$ $\Omega$ is an open domain in $\mathbf R^N$, $1<m<N$, $m^\ast-1=\dfrac{Nm}{N-m}-1$ is the critical exponent, and $f(x,u)$ has a growth exponent less than the critical one.
Theorems on the existence of a nontrivial solution of this problem is the space
$\mathring W^{1,m}(\Omega)$ and spaces of more regular functions are proved under appropriate assumptions.
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