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

Abstract

This paper deals with the existence and uniqueness results for a class of non-coercive Dirichlet elliptic problems whose model example is

$$\begin{aligned} \left\{ \begin{aligned}&-\textrm{div}\Big (\frac{1}{(m-u)^\beta } (1+|u|)^q |\nabla u|^{p-2}\nabla u+c(x)|u|^{p-2}u \sin (u-m)\Big )+g(u)=f\ \ \textrm{in}\ \Omega , \\&u=0\ \ \textrm{on}\ {\partial \Omega },\\ \end{aligned} \right. \end{aligned}$$

where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N (N\ge 2)\), \(1< p < N\), \(m>0\), \(0< \beta <1\), \(q>0\), |c| belongs to \(L^{\frac{N}{p-1}}(\Omega )\) and g is a continuous function in \({\mathbb {R}}\) which satisfies the sign condition and the data f belongs to \(L^1(\Omega )\).