Doubly critical mixed-order quasilinear equations: existence, multiplicity and regularity

Abstract

We investigate a class of mixed-order quasilinear elliptic equations in \(\mathbb R^N\) driven by the \(q\)-biharmonic and \(p\)-Laplacian operators, \[ \Delta_q^2u-\Delta_pu = \lambda\frac{|u|^{r_\sigma-2}u}{|x|^\sigma} +|u|^{p^*-2}u +|u|^{q^{**}-2}u, \] where \(1<q<N/2\), \(1<p<q^*\), \(p\ne q\), \(q\ne p^*\), and the weighted exponent is determined by the natural scaling of the equation. We establish compact weighted embeddings compatible with this mixed scaling and develop the corresponding variational framework. As an application, we prove existence and multiplicity of nontrivial solutions under suitable assumptions on the parameters. We also establish a local regularity result for the more general equation $\Delta_q^2u-\Delta_pu=f(x,u),$ where \(f\) is a Carath\'eodory function with local critical growth. By combining nonlinear potential estimates, a regularity-lifting argument, and a finite bootstrap for an associated second-order system, we obtain higher local regularity of both \(u\) and the nonlinear flux \(|\Delta u|^{q-2}\Delta u\). This regularity result, which is also of independent interest, enables us to derive a Pohozaev-type identity for the equations under consideration.