On anisotropic singular elliptic equations with a Finsler p(·)-Laplacian and variable exponents

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Abstract

This paper investigated strongly nonlinear anisotropic elliptic problems with singular terms in variable exponent Sobolev spaces. The problem involves the Finsler p(·)-Laplacian operator $\Delta_{p(\cdot)}^{H} u=$ $\operatorname{div}\left(H^{p(\cdot)-1}(\nabla u) \nabla H(\nabla u)\right),$, where H is a uniformly convex Finsler norm. We considered the differential inclusion

$\zeta(u)-\Delta_{p(\cdot)}^{H} u+G(x, u, \nabla u) \ni \frac{f}{u^{\gamma}} \quad \text { in } \Omega,$

with zero Dirichlet boundary conditions. Here ξ is a maximal monotone operator, G satisfies natural growth conditions, $f \in L^{\infty}(\Omega)$ is nonnegative, and 0 < Υ ≤ 1. Under log-Hölder continuity of p(·) and uniform convexity of H, we established existence of weak solutions via an approximation scheme combining truncation, Yosida regularization, and energy estimates. Key innovations include anisotropic monotonicity estimates, $L^{\infty}$ bounds, strict positivity, and strong convergence of gradients. Our results unify and extend previous work to encompass variable exponents, anisotropy, multivalued operators, and singular nonlinearities.

Keywords

anisotropic operators / Finsler p-Laplacian / singular nonlinearities / variable exponent Sobolev spaces / Yosida approximation

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