This paper investigated strongly nonlinear anisotropic elliptic problems with singular terms in variable exponent Sobolev spaces. The problem involves the Finsler $ p(\cdot) $-Laplacian operator $ \Delta_{p(\cdot)}^H u = \operatorname{div}(H^{p(\cdot)-1}(\nabla u)\nabla H(\nabla u)) $, 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 $ \zeta $ is a maximal monotone operator, $ G $ satisfies natural growth conditions, $ f \in L^\infty(\Omega) $ is nonnegative, and $ 0 < \gamma \leq 1 $. Under log-Hölder continuity of $ p(\cdot) $ 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.
Citation: Jiabin Zuo, Mohamed Bahadi, Youssef Akdim, Yue Wu. On anisotropic singular elliptic equations with a Finsler $ p(\cdot) $-Laplacian and variable exponents[J]. Networks and Heterogeneous Media, 2026, 21(6): 2196-2222. doi: 10.3934/nhm.2026082
This paper investigated strongly nonlinear anisotropic elliptic problems with singular terms in variable exponent Sobolev spaces. The problem involves the Finsler $ p(\cdot) $-Laplacian operator $ \Delta_{p(\cdot)}^H u = \operatorname{div}(H^{p(\cdot)-1}(\nabla u)\nabla H(\nabla u)) $, 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 $ \zeta $ is a maximal monotone operator, $ G $ satisfies natural growth conditions, $ f \in L^\infty(\Omega) $ is nonnegative, and $ 0 < \gamma \leq 1 $. Under log-Hölder continuity of $ p(\cdot) $ 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.
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Jiabin Zuo, Mohamed Bahadi, Youssef Akdim, Yue Wu. On anisotropic singular elliptic equations with a Finsler $ p(\cdot) $-Laplacian and variable exponents[J]. Networks and Heterogeneous Media, 2026, 21(6): 2196-2222. doi: 10.3934/nhm.2026082
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