In this talk, we investigate the existence of infinitely many non-constant weak solutions to the quasilinear elliptic equation
\[ -\operatorname{div}(A(x,u)∇ u) + \frac{1}{2} D_u A(x,u)∇ u \cdot ∇ u = g(x,u) -λ u, \quad u∈ H^1(Ω), \] for every \( λ ∈ \mathbb{R} \), where \(Ω \) is a bounded domain with a Lipschitz continuous boundary.

Our approach is variational; however, the associated energy functional is differentiable only along directions \( v ∈ H^1(Ω) ∩ L^∞(Ω) \).

We discuss the notion of weak slope and the critical point theory for continuous functionals, and we apply these concepts to our problem to establish the existence of multiple solutions.