Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2026
Volume
Vol. 46
Number
No. 3
Description
Journal Volume
Opuscula Mathematica
Vol. 46 (2026)
Projects
Pages
Articles
A priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions
(Wydawnictwa AGH, 2026) Corrêa, Francisco J.S.A.; Figueiredo, Giovany M.; Morbach, Joelma
In this paper, we establish a priori estimates and existence of positive solutions for elliptic problems under integral Neumann boundary conditions.
Comparison theorems for property (B) of the third-order differential equations with deviating arguments
(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.
On mixed local-nonlocal Sobolev-type inequalities and their connection with singular equations in the Heisenberg group
(Wydawnictwa AGH, 2026) Garain, Prashanta
In this work, we establish a mixed local-nonlocal Sobolev-type inequality in the Heisenberg group and demonstrate that its extremals coincide with solutions to the corresponding mixed local-nonlocal singular $p$-Laplace equations. We further show that these inequalities serve as a necessary and sufficient condition for the existence of weak solutions to the associated singular problems. Notably, the same characterization remains valid in both the purely local and purely nonlocal settings. Our results thus provide a unified framework linking the existence theory for singular equations across local, nonlocal, and mixed regimes.
On a relation between growth estimates and Harnack inequalities for quasilinear elliptic equations with nonlinear lower order terms
(Wydawnictwa AGH, 2026) Hirata, Kentaro
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 $\mathbb{R}^N$.
Existence of solutions for a doubly critical Schrödinger-Poisson system on the first Heisenberg group
(Wydawnictwa AGH, 2026) Ma, Xueyan; Shi, Shaoyun; Song, Yueqiang
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.