Towards existence theorems to affine p-Laplace equations via variational approach.

The present work deals with theory of critical points to the energy functional on W 0 1 , p (Ω) defined by Φ A (u) = 1 p E p , Ω p (u) - ∫ Ω F (x , u) d x , where E p , Ω p stands for the affine p-energy introduced for p > 1 by Lutwak et al. (J Differ Geom 62:17–38, 2002). Its development is inspired in the study of solutions of elliptic equations involving the affine p-Laplace non-local operator Δ p A introduced recently by Haddad et al. (Adv Math 386:107808, 2021). New results on regularity of E p , Ω p and (S +) compactness property associated to Δ p A are established. The latter is fundamental on the discussion of Palais–Smale compactness for Φ A when affine mountain-pass and coercive geometries are considered. The mountain-pass case is quite intricate, being addressed by means of asymptotic analysis as ε → 0 of corresponding critical points of the one-parameter perturbation Φ A ε (u) = Φ A (u) + ε ‖ u ‖ W 0 1 , p (Ω) . [ABSTRACT FROM AUTHOR]