Elliptic problems involving a diffusion operator of vectorial $p\&q$-Laplacian type

Abstract

We study a broad class of anisotropic Dirichlet problems driven by the nonhomogeneous operator \[ \mathcal L_Au=-\sum_{i=1}^N\partial_i\!\left( a_i(|\partial_i u|^{p_i})|\partial_i u|^{p_i-2} \partial_i u\right), \] where the functions $a_i$ satisfy structural growth and monotonicity conditions of $(p_i,q_i)$-type. This framework contains, among other examples, the anisotropic $\vec p$-Laplacian, the orthotropic $p\&q$-Laplacian, and anisotropic operators combining several directional growth regimes. We first establish a comparison principle adapted to this lack of homogeneity. For power nonlinearities, including the singular range, we then obtain nonexistence, existence, singular uniqueness, comparison alternatives, and multiplicity results according to the position of the reaction exponent relative to the anisotropic growth and critical exponents. In particular, an intermediate power regime exhibits nonexistence for small positive parameters and two distinct nontrivial solutions for large parameters without additional restrictions on the smallest effective exponent. At partial resonance, $r=p_1<p_N$, we identify an exact positive parameter threshold: there are no nontrivial solutions at or below it, whereas each parameter above it admits infinitely many pairs of solutions converging strongly to zero. We also treat concave--convex reactions with subcritical or critical growth and prove the existence of nontrivial nonnegative weak solutions in the corresponding parameter ranges. The arguments combine a convexity method of Díaz--Saa type, anisotropic Sobolev embeddings, direct minimization, Ekeland's variational principle, mountain-pass techniques, and concentration--compactness.