Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

Abstract

In this paper we prove existence results and asymptotic behavior for strong solutions u ∈ W 2,2 loc (Ω) of the nonlinear elliptic problem (P) −∆ H u + H(∇u) q + λu = f in Ω, u → +∞ on ∂Ω, where H is a suitable norm of R n , Ω ⊂ R n is a bounded domain, ∆ H is the Finsler Laplacian, 1 < q ≤ 2, λ > 0, and f is a suitable function in L ∞ loc. Furthermore, we are interested in the behavior of the solutions when λ → 0 + , studying the so-called ergodic problem associated to (P). A key role in order to study the ergodic problem will be played by local gradient estimates for (P).

Asymptotic behavior of least energy solutions to the Finsler Lane-Emden problem with large exponents

“…If we assume that Hess(H q (ξ)) is positive definite on R N \ {0}, Q q becomes a uniformly elliptic operator locally. The Finsler q-Laplacian has been widely investigated in literature by many authors in different settings, see [2], [5], [8], [9], [10], [12], [13], [14], [17], [20], [24], [35] and the references therein.…”

Finsler Hardy inequalities

“…The Finsler Laplacian has been widely investigated in literature and its notion goes back to the work of G. Wulff, who considered it to describe the theory of crystals. Many other authors developed the related theory in several settings, considering both analytic and geometric points of view, see ([4], [2], [12], [13], [14], [18], [20], [21] and references therein).…”

On the anisotropic Moser-Trudinger inequality for unbounded domains in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^{n} $\end{document}</tex-math></inline-formula>

“…We call this nonlinear operator as Finsler-Laplacian. This operator Q n was studied by many mathematicians, see [2,3,10,27,36,37,38,40] and the references therein.…”