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 \(W^{1,p}_0(\Omega )\) $W_0^{1,p}(\mathrm{\Omega })$ defined by \(\begin{aligned} \Phi _\mathcal{A}(u) = \frac{1}{p} \mathcal{E}^p_{p,\Omega }(u) - \int _\Omega F(x,u)\,dx, \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Phi }}_{\mathcal{A}}(u)=\frac{1}{p}{\mathcal{E}}_{p,\mathrm{\Omega }}^p(u)-{\int }_{\mathrm{\Omega }}F(x,u)dx,\end{array}$$ where \(\mathcal{E}^p_{p,\Omega }\) ${\mathcal{E}}_{p,\mathrm{\Omega }}^p$ stands for the affine p-energy introduced for \(p > 1\) $p>1$ 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 \(\Delta _p^\mathcal{A}\) ${\mathrm{\Delta }}_p^{\mathcal{A}}$ introduced recently by Haddad et al. (Adv Math 386:107808, 2021). New results on regularity of \(\mathcal{E}^p_{p,\Omega }\) ${\mathcal{E}}_{p,\mathrm{\Omega }}^p$ and \((S_+)\) $(S_{+})$ compactness property associated to \(\Delta _p^\mathcal{A}\) ${\mathrm{\Delta }}_p^{\mathcal{A}}$ are established. The latter is fundamental on the discussion of Palais–Smale compactness for \(\Phi _\mathcal{A}\) ${\mathrm{\Phi }}_{\mathcal{A}}$ when affine mountain-pass and coercive geometries are considered. The mountain-pass case is quite intricate, being addressed by means of asymptotic analysis as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ of corresponding critical points of the one-parameter perturbation \(\Phi ^\varepsilon _\mathcal{A}(u) = \Phi _\mathcal{A}(u) + \varepsilon \Vert u \Vert _{W_0^{1,p}(\Omega )}\) ${\mathrm{\Phi }}_{\mathcal{A}}^{\epsilon }(u)={\mathrm{\Phi }}_{\mathcal{A}}(u)+\epsilon {\Vert u\Vert }_{W_0^{1,p}(\mathrm{\Omega })}$ .