This work is devoted to the study of a class of Schrödinger-Poisson system with doubly critical growth on the first Heisenberg group. Utilizing the concentration-compactness principle associated with classical Sobolev space on the Heisenberg group and mountain pass theorem, we prove that the system admits multiple nontrivial solutions.
We investigate a relation between the Harnack inequalities and the (a priori) growth estimates for positive solutions of quasilinear elliptic equations with nonlinear terms involving the solution and its gradient in an arbitrary domain in $\mathbb{R}^N$.
(Wydawnictwa AGH, 2026) Džurina, Jozef; Baculíková, Blanka
The aim of this paper is to introduce a new comparison theorem (in both delayed and advanced cases) that allows us to investigate the properties of third-order differential equations with quasi-derivatives $(r_{1}(t)(r_{2}(t)y'(t))')'-p(t)y(\tau(t))=0$ using the following simpler differential equations $(r(t)(r(t)z'(t))')'-p(t)z(\tau(t))=0$ and $y'''(t)-q(t)y(\sigma(t))=0.$ The obtained comparison principles allow for the immediate transcription of the oscillatory results known for the simpler equations into studied equation with quasi-derivatives. The progress achieved will be illustrated through several examples.
