Least energy solutions for a quasilinear Neumann problem

We study the existence of nonconstant weak solutions and the qualitative behavior of the least energy level for a class of quasilinear elliptic problems with Neumann boundary conditions of the form $$-\sum _{i,j=1}^N{D}_{x_j}(a_{ij}(x,u)D_{x_i}u)+\frac{1}{2}\sum _{i,j=1}^N{D}_s{a}_{ij}(x,u)D_{x_i}u{D}_{x_j}u=g(u)-\lambda u\text{in }\mathrm{\Omega }\mathrm{.}$$ Since the associated energy functional is neither of class $C^1$ nor locally Lipschitz, the classical Nehari manifold method cannot be applied directly. We therefore rely on variational techniques combined with nonsmooth critical point theory. For $\lambda >0$, we prove the existence of least energy solutions and show that, under suitable assumptions, they are nonconstant for large $\lambda$ and constant for sufficiently small $\lambda$. Moreover, the least energy level tends to $+\mathrm{\infty }$ as $\lambda \to +\mathrm{\infty }$. In the indefinite regime, a key ingredient is a spectral decomposition of $H^1(\mathrm{\Omega })$ associated with the linear operator $${\mathcal{L}}_{0}u:=-\sum _{i,j=1}^N{D}_{x_j}(a_{ij}(x,0)D_{x_i}u)\mathrm{.}$$