Quasilinear Equations with Neumann Boundary Conditions

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation $$\begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) - \lambda u & \text{in } \Omega \\ A(x,u)\nabla u \cdot \eta = 0 & \text{on } \partial \Omega \end{cases}$$ where $\lambda \in \mathbb{R}$, $\Omega$ is a bounded lipschitz domain, $\eta$ is the outward normal to the boundary $\partial \Omega$, and $g(x,u)$ is a Carath\'eodory function that satisfies a general subcritical (and superlinear) growth condition. We also prove that any weak solution is bounded under a stronger growth assumption.