Solvability of some elliptic problems with critical exponent of nonlinearity

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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