Year: 2026
Type: article
Abstract: We are concerned with the existence of solution of the problem where $$\Delta ^H_pu=\hbox {div }(a(\nabla u))$$ Δ p H u = div ( a ( ∇ u ) ) , with $$a(\xi )=H^{p-1}(\xi )\nabla H(\xi ),\, \xi \in \mathbb {R}^N,$$ a ( ξ ) = H p - 1 ( ξ ) ∇ H ( ξ ) , ξ ∈ R N , $$N\geqslant 3,$$ N ⩾ 3 , is the anisotropic p -Laplacian with $$1 1 < p < N , $$\lambda >0$$ λ > 0 is a parameter, and $$p< q p < q < p ∗ = p N / ( N - p ) . Further, $$\Omega $$ Ω is a $$C^1$$ C 1 bounded domain inside a convex open cone. To succeed with a variational approach, where the strong convergence of a bounded (PS) subsequence needs to be proved, one has to deal with anisotropic norms in the absence of a Tartar’s type inequality, unlike the isotropic p -Laplace case. This is overcome by proving the a.e. convergence of its gradients. Furthermore, the solution of ( P ) is shown to belong to $$C^{1,\alpha }(\Omega )$$ C 1 , α ( Ω ) from classical elliptic regularity theory, and is positive from a Harnack inequality, since any solution of ( P ) is bounded. This in turn is a consequence of a resu
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W7129604412