We consider the following problem:

\begin{alignat*}{2} -\text{div}(A(x,u)\nabla u)&=u^s+f(x) & \quad &\text{in }\varOmega, \\ u(x)&\ge0 & & \text{in }\varOmega, \\ u(x)&=0 & & \text{on }\p\varOmega, \end{alignat*}

where $\varOmega$ is an open bounded subset of $\mathbb{R}^N$, $N\ge3$, and

$$ A:\varOmega\times\mathbb{R}\rightarrow M_{N\times N} $$

is an elliptic matrix such that when $u\to\infty$ is non-coercive.