Quasilinear Equations with Neumann Boundary Conditions

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