Opuscula MathematicaA list of articles of the latest volume. The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics.
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Concentration-compactness results for systems in the Heisenberg groupIn this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895–922] on some variants of the concentration-compactness principle in bounded PS domains \(\Omega\) of the Heisenberg group \(\mathbb{H}^n\). The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4009.pdf
Concentration-compactness results for systems in the Heisenberg groupPatrizia PucciLetizia TemperiniHeisenberg group, concentration-compactness, critical exponents, Hardy termsdoi:10.7494/OpMath.2020.40.1.151Opuscula Math. 40, no. 1 (2020), 151-163, https://doi.org/10.7494/OpMath.2020.40.1.151Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.151https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4009.pdf401151163 Title: Concentration-compactness results for systems in the Heisenberg group.

Author(s): Patrizia Pucci, Letizia Temperini.

Abstract: In this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895–922] on some variants of the concentration-compactness principle in bounded PS domains \(\Omega\) of the Heisenberg group \(\mathbb{H}^n\). The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms. Keywords: Heisenberg group, concentration-compactness, critical exponents, Hardy terms. Mathematics Subject Classification: 22E30, 35B33, 35J50, 58E30, 35H05, 35A23. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 151-163, https://doi.org/10.7494/OpMath.2020.40.1.151.

]]>A multiplicity theorem for parametric superlinear (p,q)-equationsWe consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4008.pdf
A multiplicity theorem for parametric superlinear (p,q)-equationsFlorin-Iulian OneteNikolaos S. PapageorgiouCalogero Vetrosuperlinear reaction, constant sign and nodal solutions, extremal solutions, nonlinear regularity, nonlinear maximum principle, critical groupsdoi:10.7494/OpMath.2020.40.1.131Opuscula Math. 40, no. 1 (2020), 131-149, https://doi.org/10.7494/OpMath.2020.40.1.131Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.131https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4008.pdf401131149 Title: A multiplicity theorem for parametric superlinear (p,q)-equations.

Author(s): Florin-Iulian Onete, Nikolaos S. Papageorgiou, Calogero Vetro.

Abstract: We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information. Keywords: superlinear reaction, constant sign and nodal solutions, extremal solutions, nonlinear regularity, nonlinear maximum principle, critical groups. Mathematics Subject Classification: 35J20, 35J60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 131-149, https://doi.org/10.7494/OpMath.2020.40.1.131.

]]>Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearityIn this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4007.pdf
Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearityWei LianMd Salik AhmedRunzhang Xuglobal existence, blow-up, logarithmic and polynomial nonlinearity, potential welldoi:10.7494/OpMath.2020.40.1.111Opuscula Math. 40, no. 1 (2020), 111-130, https://doi.org/10.7494/OpMath.2020.40.1.111Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.111https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4007.pdf401111130 Title: Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity.

Author(s): Wei Lian, Md Salik Ahmed, Runzhang Xu.

Abstract: In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity. Keywords: global existence, blow-up, logarithmic and polynomial nonlinearity, potential well. Mathematics Subject Classification: 35L71, 35L20, 35L05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 111-130, https://doi.org/10.7494/OpMath.2020.40.1.111.

]]>Fractional p&q-Laplacian problems with potentials vanishing at infinityIn this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p\&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4006.pdf
Fractional p&q-Laplacian problems with potentials vanishing at infinityTeresa Iserniafractional \(p\&q\)-Laplacian, vanishing potentials, ground state solutiondoi:10.7494/OpMath.2020.40.1.93Opuscula Math. 40, no. 1 (2020), 93-110, https://doi.org/10.7494/OpMath.2020.40.1.93Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.93https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4006.pdf40193110 Title: Fractional p&q-Laplacian problems with potentials vanishing at infinity.

Author(s): Teresa Isernia.

Abstract: In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p\&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator. Keywords: fractional \(p\&q\)-Laplacian, vanishing potentials, ground state solution. Mathematics Subject Classification: 35A15, 35J60, 35R11, 45G05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 93-110, https://doi.org/10.7494/OpMath.2020.40.1.93.

]]>Nonhomogeneous equations with critical exponential growth and lack of compactnessWe study the existence and multiplicity of positive solutions for the following class of quasilinear problems \[-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,\] where \(\epsilon\) is a positive parameter. We assume that \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous potential and \(f:\mathbb{R}\to\mathbb{R}\) is a smooth reaction term with critical exponential growth.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4005.pdf
Nonhomogeneous equations with critical exponential growth and lack of compactnessGiovany M. FigueiredoVicenţiu D. Rădulescuexponential critical growth, quasilinear equation, Trudinger-Moser inequality, Moser iterationdoi:10.7494/OpMath.2020.40.1.71Opuscula Math. 40, no. 1 (2020), 71-92, https://doi.org/10.7494/OpMath.2020.40.1.71Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.71https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4005.pdf4017192 Title: Nonhomogeneous equations with critical exponential growth and lack of compactness.

Author(s): Giovany M. Figueiredo, Vicenţiu D. Rădulescu.

Abstract: We study the existence and multiplicity of positive solutions for the following class of quasilinear problems \[-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,\] where \(\epsilon\) is a positive parameter. We assume that \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous potential and \(f:\mathbb{R}\to\mathbb{R}\) is a smooth reaction term with critical exponential growth. Keywords: exponential critical growth, quasilinear equation, Trudinger-Moser inequality, Moser iteration. Mathematics Subject Classification: 35J62, 35A15, 35B30, 35B33, 58E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 71-92, https://doi.org/10.7494/OpMath.2020.40.1.71.

]]>On the regularity of solution to the time-dependent p-Stokes systemIn this paper we consider the time evolutionary \(p\)-Stokes problem in a smooth and bounded domain. This system models the unsteady motion or certain non-Newtonian incompressible fluids in the regime of slow motions, when the convective term is negligible. We prove results of space/time regularity, showing that first-order time-derivatives and second-order space-derivatives of the velocity and first-order space-derivatives of the pressure belong to rather natural Lebesgue spaces.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4004.pdf
On the regularity of solution to the time-dependent p-Stokes systemLuigi C. BerselliMichael Růžičkaregularity, evolution problem, \(p\)-Stokesdoi:10.7494/OpMath.2020.40.1.49Opuscula Math. 40, no. 1 (2020), 49-69, https://doi.org/10.7494/OpMath.2020.40.1.49Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.49https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4004.pdf4014969 Title: On the regularity of solution to the time-dependent p-Stokes system.

Author(s): Luigi C. Berselli, Michael Růžička.

Abstract: In this paper we consider the time evolutionary \(p\)-Stokes problem in a smooth and bounded domain. This system models the unsteady motion or certain non-Newtonian incompressible fluids in the regime of slow motions, when the convective term is negligible. We prove results of space/time regularity, showing that first-order time-derivatives and second-order space-derivatives of the velocity and first-order space-derivatives of the pressure belong to rather natural Lebesgue spaces. Keywords: regularity, evolution problem, \(p\)-Stokes. Mathematics Subject Classification: 76D03, 35Q35, 76A05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 49-69, https://doi.org/10.7494/OpMath.2020.40.1.49.

]]>On solvability of elliptic boundary value problems via global invertibilityIn this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4003.pdf
On solvability of elliptic boundary value problems via global invertibilityMichał BełdzińskiMarek Galewskidiffeomorphism, Dirichlet conditions, Laplace operator, Neumann conditions, uniquenessdoi:10.7494/OpMath.2020.40.1.37Opuscula Math. 40, no. 1 (2020), 37-47, https://doi.org/10.7494/OpMath.2020.40.1.37Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.37https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4003.pdf4013747 Title: On solvability of elliptic boundary value problems via global invertibility.

Author(s): Michał Bełdziński, Marek Galewski.

Abstract: In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions. Keywords: diffeomorphism, Dirichlet conditions, Laplace operator, Neumann conditions, uniqueness. Mathematics Subject Classification: 35J60, 46T20, 47H30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 37-47, https://doi.org/10.7494/OpMath.2020.40.1.37.

]]>Some multiplicity results of homoclinic solutions for second order Hamiltonian systems We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4002.pdf
Some multiplicity results of homoclinic solutions for second order Hamiltonian systemsSara BarileAddolorata Salvatoresecond order Hamiltonian systems, homoclinic solutions, variational methods, compact embeddingsdoi:10.7494/OpMath.2020.40.1.21Opuscula Math. 40, no. 1 (2020), 21-36, https://doi.org/10.7494/OpMath.2020.40.1.21Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.21https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4002.pdf4012136 Title: Some multiplicity results of homoclinic solutions for second order Hamiltonian systems.

Author(s): Sara Barile, Addolorata Salvatore.

Abstract: We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough. Keywords: second order Hamiltonian systems, homoclinic solutions, variational methods, compact embeddings. Mathematics Subject Classification: 34C37, 58E05, 70H05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 21-36, https://doi.org/10.7494/OpMath.2020.40.1.21.

]]>On some convergence results for fractional periodic Sobolev spacesIn this note we extend the well-known limiting formulas due to Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova, to the setting of fractional Sobolev spaces on the torus. We also give a \(\Gamma\)-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series.
https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4001.pdf
On some convergence results for fractional periodic Sobolev spacesVincenzo Ambrosiofractional periodic Sobolev spaces, Fourier series, \(\Gamma\)-convergencedoi:10.7494/OpMath.2020.40.1.5Opuscula Math. 40, no. 1 (2020), 5-20, https://doi.org/10.7494/OpMath.2020.40.1.5Copyright AGH University of Science and Technology Press, Krakow 2020Opuscula Mathematica20202020https://doi.org/10.7494/OpMath.2020.40.1.5https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4001.pdf401520 Title: On some convergence results for fractional periodic Sobolev spaces.

Author(s): Vincenzo Ambrosio.

Abstract: In this note we extend the well-known limiting formulas due to Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova, to the setting of fractional Sobolev spaces on the torus. We also give a \(\Gamma\)-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series. Keywords: fractional periodic Sobolev spaces, Fourier series, \(\Gamma\)-convergence. Mathematics Subject Classification: 42B05, 46E35, 49J45. Journal: Opuscula Mathematica. Citation: Opuscula Math. 40, no. 1 (2020), 5-20, https://doi.org/10.7494/OpMath.2020.40.1.5.