Existence and uniqueness results for an elliptic equation with blowing-up coefficient and lower order term

This paper deals with the existence and uniqueness results for a class of non-coercive Dirichlet elliptic problems whose model example is \(\begin{aligned} \left\{ \begin{aligned}&-\textrm{div}\Big (\frac{1}{(m-u)^\beta } (1+|u|)^q |\nabla u|^{p-2}\nabla u+c(x)|u|^{p-2}u \sin (u-m)\Big )+g(u)=f\ \ \textrm{in}\ \Omega , \\&u=0\ \ \textrm{on}\ {\partial \Omega },\\ \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& -\text{div}(\frac{1}{{(m-u)}^{\beta }}{(1+|u|)}^q{|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u+c(x){|u|}^{p-2}usin(u-m))+g(u)=f\text{in}\mathrm{\Omega },\hfill \\ \hfill & u=0\text{on}\partial \mathrm{\Omega },\hfill \\ \hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded open subset of \({\mathbb {R}}^N (N\ge 2)\) ${\mathbb{R}}^N(N\ge 2)$ , \(1< p < N\) $1<p<N$ , \(m>0\) $m>0$ , \(0< \beta <1\) $0<\beta <1$ , \(q>0\) $q>0$ , |c| belongs to \(L^{\frac{N}{p-1}}(\Omega )\) $L^{\frac{N}{p-1}}(\mathrm{\Omega })$ and g is a continuous function in \({\mathbb {R}}\) $\mathbb{R}$ which satisfies the sign condition and the data f belongs to \(L^1(\Omega )\) $L^1(\mathrm{\Omega })$ .