Solving Strongly Sublinear p(x)-Laplacian Dirichlet Problems: Existence via Fixed-Point Theorems

We study the existence of weak solutions to the nonlinear elliptic problem −Δp(x) u = λ|u|s(x)−2 u + f(x, u,∇u) in a bounded domain Ω ⊂ RN with smooth boundary ∂Ω, under homogeneous Dirichlet conditions. The equation features a variable exponent p(x) in the p(x)-Laplacian operator, an eigenvalue term λ|u|s(x)−2 u, and a Carathéodory perturbation f with sublinear growth, depending on both u and ∇u. Using a topological approach based on fixed-point theorems, we establish the existence of weak solutions to the problem.