On Strongly Nonlinear Elliptic Equations with Variable Exponents Without Sign Assumption

We investigate the existence of solutions for a broad class of strongly nonlinear elliptic problems governed by the inclusion \[ \beta(u) + A(u) + H(x,u,\nabla u) \ni f, \] where the operator \( A \) is of Leray--Lions type and acts between the variable exponent Sobolev space \( W_0^{1,p(x)}(\Omega) \) and its dual. The multivalued mapping \( \beta \) is maximal monotone such that \( 0 \in \beta(0) \), while the nonlinearity \( H(x,s,\eta) \) is allowed to exhibit strong growth in its arguments but is not assumed to satisfy any sign condition with respect to \( s \). The source term \( f \) is supposed to lie in \( L^\infty(\Omega) \). Our analysis is carried out within the framework of variable exponent Sobolev spaces, which reflect the nonstandard growth behavior inherent to the problem.