We analyze the structure of the set of solutions to the following class of boundary value problems \[\begin{cases}-\div(A(x)Du)=c_\lambda(x)u+( M(x)Du,Du)+h(x)\\u\in H_0^1(\Omega)\cup L^\infty(\Omega)\end{cases}\tag{$P_\lambda$}\label{$P_lambda$}\] where $\Omega\subset\mathbb{R}^n$, $n\geq 3$ is a bounded domain with boundary $\partial\Omega$ of class $C^{1,D}$. We assume that $c,h \in L^p(\Omega)$ for some $p>n$, where $c^{\pm} \geq 0$ are such that $c_\lambda(x):=\lambda c^+(x)-c^-(x)$ for a parameter $\lambda\in\mathbb{R}$, $A(x)$ is a uniformly positive bounded measurable matrix and $M(x)$ is a positive bounded matrix. Under suitable assumptions, we describe the continuum of solutions of the problem ($P_\lambda$) and also its bifurcation points proving existence and uniqueness results in the coercive case $(\lambda \leq 0)$ and multiplicity results in the non-coercive case $(\lambda > 0)$.