Least energy solutions for affine p-Laplace equations involving subcritical and critical nonlinearities

Abstract

The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 𝒜 {\Delta_{p}^{\cal A}} , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p {L^{p}} energy ℰ p , Ω {{\cal E}_{p,\Omega}} from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p L_{p} Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1 {C^{1}} solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω {{\cal E}_{p,\Omega}} and by the comparison ℰ p , Ω ⁢ ( u ) ≤ ∥ u ∥ W 0 1 , p ⁢ ( Ω ) {{\cal E}_{p,\Omega}(u)\leq\|u\|_{W^{1,p}_{0}(\Omega)}} generally strict.