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List of topics:
 - Applications to Electronics and Fiber Optics (2)
 - Applications to Industry (1)
 - Applications to Medicine (3)
 - Applications to Nanoscience and Comp. Chemistry (1)
 - Educational and Multimedia Software (4)
 - Integral Equations and Linear Functional Analysis (2)
 - Mathematical Knowledge and Preservation (3)
 - Noncommutative Geometry (3)
 - Nonlinear Control Theory (10)
 - Partial Differential Equations (31)

Applications to Electronics and Fiber Optics (2)

Stability analysis of Raman propagation equations (with B. Neto, M.M. Rodrigues, P.S. Andre), IEEE – Conference Proceedings, 11th International Conference Transparent Optical Networks, 2009 – ICTON ’09 (IEEE – Conference Proceedings), 1-4, Ponta Delgada – Açores, 2009.
(myid-77)

Stability analysis of Raman propagation equations of three and higher dimensions (with B. Neto, M.M. Rodrigues, R.M. Morais, A.N. Pinto, P.S. Andre), Optical Networks – SEON, 2008.
(myid-76)

Applications to Industry (1)

Solving huge size instances of optimal diversity management problem (with A. Agra, D. Cardoso, O. Cerdeira, M. Miranda), J. Mathematical Sciences 161:6(2009), 956-960.

Extra info: (P008)

Applications to Medicine (3)

The effect of immigrant communities coming from higher incidence tuberculosis regions to a host country (with C.J. Silva, D.F.M. Torres), pp.19, preprint.

ABSTRACT: W15-08 We introduce a new tuberculosis (TB) mathematical model, with 25 state-space variables where 15 are evolution disease states (EDSs), which generalises previous models and takes into account the flux of populations between a high incidence TB country (A) and a community (G) with high percentage of people from (A), plus the rest of the population (C) of a host country (B) with low TB incidence. Contrary to some beliefs, related to the fact that agglomerations of individuals increase proportionally to the disease spread, analysis of the model shows that the existence of communities are simultaneously beneficial for the TB control from a global and regional viewpoint. There is an optimal ratio for the distribution of individuals in (C) versus (G), which minimizes the reproduction number R_0. Such value does not give the minimal total number of infected individuals in all (B), since such is attained when the community (G) is completely isolated (theoretical scenario). Sensitivity analysis and curve fitting on R_0 and on EDSs are pursuit in order to understand the TB effects in the global statistics, by measuring the variability of the relevant parameters that account for the existence of (G), composed of individuals coming from a high incidence area, and the (seasonal) flux between (A) and (B). We also show that the TB transmission rate \beta does not act linearly on R_0, as is common in compartment models where system feedback or group interactions do not occur. Further, we find the most important parameters for the increase of each EDS. The model and techniques proposed are applied to a case-study with concrete parameters, which model the situation of Angola (A) and Portugal (B), in order to show its relevance and meaningfulness.
Extra info: (P103)

Quantitative assessment of retinal structures and lesions (with H. Araújo, J. Cunha-Vaz, L. Ribeiro, D. Torres), in: Proceedings of RecPad95, 6.5.1-6.5.6, Aveiro, Portugal, 1995.
(myid-14)

Aquisição directa, processamento computorizado e tridimensionalização de imagens da área avascular central da retina humana (with H.J. Araújo, D. Torres), B.Sc. Diploma Thesis, Faculty of Science and Technology of the University of Coimbra and Institute of Biomedical Research on Light and Image, Portugal, Jul/1994 (JNICT Reference: PO435.03.01).

Extra info: (P013)

Applications to Nanoscience and Comp. Chemistry (1)

The origin of the LCST on the liquid−liquid equilibrium of thiophene with ionic liquids (with M.L.S. Batista, L.I.N. Tomé, C.M.S.S. Neves,
J.R.B. Gomes, J.A.P. Coutinho), J. Phys. Chem. B, 116:20(2012), 5985–5992.

ABSTRACT: normal.img-000Mixtures of thiophene with two ionic liquids, namely, [C_4C_1im][SCN] and [C_4C_1im][NTf_2] were chosen as prototypes of systems presenting lower critical solution temperature (LCST) and upper critical solution temperature (UCST) behavior, respectively. This distinct behavior is due to different interactions between the constituting species which are investigated here by means of experimental and computational studies. Experimentally, density measurements were conducted to assess the excess molar volumes and ^{1}H and ^{13}C NMR spectroscopies were used to obtain the corresponding nuclear chemical shifts with respect to those measured for the pure ionic liquids. Computationally, molecular dynamics simulations were performed to analyze the radial distribution neighborhoods of each species. Negative values of excess molar volumes and strong positive chemical shift deviations for [C_4C_1im][SCN] systems, along with results obtained from MD simulations, allowed the identification of specific interactions between [SCN]^- anion and the molecular solvent (thiophene), which are not observed for [NTf_2]^-. It is suggested that these specific [SCN]^--thiophene interactions are responsible for the LCST behavior observed for mixtures of thiophene with ionic liquids.

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1021/jp303187z)
Extra info: (P101/2012-05-02)

Educational and Multimedia Software (4)

The IntBook concept as an adaptative web environment (with A. Breda, T. Parreira), in: Communicating Mathematics in the Digital Era, J. Borwein et al. (eds), 253-268, A. K. Peters Ltd, 2008.
(myid-7)

Interactive, collaborative and adaptative learning tools (with A. Breda, M.M. Rodrigues), in: Proceedings of the 4th International Conference on Web Information Systems and Technology, INSTICC, José Cordeiro et al. (eds), 456-459, Funchal-Madeira, 2008.
(myid-2)

TexMat – a multimedia constructivist learning environment (with A. Breda, J. Martins), in: Proceedings of the 11th Asian Technology Conference in Mathematics (ATCM), Sung-Chi Chu et al. (eds), 12-16, Hong Kong, 2006.
(myid-25)

BioMat – a multi-mode educational game (with A. Breda), in: Proceedings of ED-MEDIA’2005, 1359-1366, Canada, 2005.
(myid-23)

Integral Equations and Linear Functional Analysis (2)

A reproducing kernel Hilbert discretization method for linear PDEs with nonlinear right-hand side, Lib. Math. 34:2(2014), 91-104.

ABSTRACT: In this paper, the main problem of study concerns to find a suitable discretization method to (numerically represent) the solution u\in\mathcal{H}, in a Hilbert space, of the general equation

(1)   \begin{equation*} L[u(x)]=f(x,u(x)),\quad x\in\Omega\subset\bkR^N, \end{equation*}

satisfying a given boundary condition (e.g. Dirichlet), where L is a linear differential (or integral) operator and f is a nonlinear function with enough regularity. The method Aveiro Discretization Method in Mathematics (ADMM), introduced by Saitoh et al. (2014; doi:10.1007/978-1-4939-1106-6 3), can deal with problem (1) when f is a function that do not depends on u. In fact, ADMM is a general method for solving by discretization, in a specific optimal sense and when a priori some data may be missing, a wide class of linear mathematical problems by using some key ideas of reproducing kernels and Tikhonov regularization theory. Here, we aim to extend the ADMM method to a more general situation where the nonlinearity may depend on u. Then, we apply the scheme to find the (optimal) discretization solution of the problem

    \[\vartheta\frac{\partial^2 u}{\partial y^2}+\alpha \frac{\partial^2 u}{\partial x^2}+\gamma u= u^3,\quad \mbox{ on } \bkR^2,\]

for arbitrary coefficient functions \vartheta, \alpha, \gamma.

Download: Preprint version (RIA |  download) / Publisher version (doi:10.14510/lm-ns.v34i2.1306)
Extra info: (P105/2014)

Existence, uniqueness and differentiability of the solution of a functional equation with general delay (with L.P. Castro), Annals of the Univ. of Cracovia XXIX(2002), 1-9.
(myid-19)

Mathematical Knowledge and Preservation (3)

Communicating Mathematics in the Digital Era, J.M. Borwein, E.M. Rocha and J.F. Rodrigues (eds), pp. 325, A.K. Peters, 2008.
(myid-63)

Disseminating and preserving the mathematical knowledge (with J.F. Rodrigues), in: Communicating Mathematics in the Digital Era, J. Borwein et al. (eds), 3-21, A. K. Peters Ltd, 2008.
(myid-3)

Comunicação da matemática na era digital (with J.F. Rodrigues), Boletim Soc. Port. Mat. 53(2005), 1-21.
(myid-24)

Noncommutative Geometry (3)

Sign changing solutions of a semilinear equation on Heisenberg group, (with J. Chen) accepted in J. Non. and Convex Anal. (2015).

ABSTRACT: This paper is concerned with the existence of multiple solutions to the semilinear equation

    \[\Delta_H\, u + |u|^{\frac{4}{Q-2}}u+\mu|\xi|^\alpha_H u=0\]

in a bounded domain of the Heisenberg group \mathbb{H}^N with Dirichlet boundary condition, where \alpha > 0 and |\xi|_H is a distance in \mathbb{H}^N. By using variational methods, we prove that this problem possesses at least one positive solution and one sign changing solution for some values of \alpha and \mu.

Download: Preprint version (RIA |  download) / Publisher version (not available yet)
Extra info: (P090/2015-06)

Existence of solutions of sub-elliptic equations on the Heisenberg group with critical growth and singularity (with J. Chen), Opuscula Math. 33:2(2013), 237-254.

ABSTRACT: We consider a class of sub-elliptic equations on the Heisenberg group \mathbb{H}^N with a Hardy type singularity and a critical nonlinear growth

    \[-\Delta_{H,p} u -\lambda\frac{|z|^p}{\rho^{2p}}|u|^{p-2}u = \frac{|z|^s}{\rho^{2s}}|u|^{p_\ast(s)-2}u\quad\mbox{ in }\mathbb{H}^N\backslash\{0\},\]

where \Delta_{H,p} is the p-sub-Laplacian with respect to a fixed bases for the corresponding Lie algebra of the Heisenberg group, (z,t)=(x,y,t)\in\mathbb{H}^N, \rho(\xi) is the distance of \xi\in\mathbb{H}^N to the origin given by the homogeneous distance

    \[d(\xi,\xi')=\left((|x-x'|^2+|y-y'|^2)^2+(t-t'-2(x,y')_{\mathbb{R}}-2(x',y)_{\mathbb{R}})^2\right)^{\frac{1}{4}},\]

Q=2N+2 is the homogeneous dimension, \mathcal{D}^{1,p}_0\left(\mathbb{H}^N\right) denotes the closure of C^\infty_0\left(\mathbb{H}^N\right) under the norm \|u\|=(\int_{\mathbb{H}^N}|\nabla_H u|^p\,d\xi)^{\frac{1}{p}} and p_\ast(s)=p(Q-s)(Q-p)^{-1} the critical exponent of embedding

    \[\mathcal{D}^{1,p}_0\left(\mathbb{H}^N\right) \hookrightarrow L^{p_\ast(s)}\left(\mathbb{H}^N\right, \frac{|z|^s}{\rho^{2s}}d\xi).\]

We prove the existence of energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252].

Download: Preprint version (RIA |  download) / Publisher version (doi:10.7494/OpMath.2013.33.2.237)
Extra info: (P080/2013) (free/open access)

A class of sub-elliptic equations on the Heisenberg group and related interpolation inequalities (with J. Chen) in Advances in Harmonic Analysis and Operator Theory, Operator Theory: Advances and Applications 229, 123–137, (2013).

ABSTRACT: We firstly prove the existence of least energy solutions to a class of sub-elliptic equations on the Heisenberg group \mathbb{H}^N of the form

    \[-\Delta_H u + u = |u|^{p-2}u\quad u\in S^{1,2}_0(\mathbb{H}^N),\]

where S^{1,2}_0(\mathbb{H}^N) is the closure of C^{\infty}_0(\mathbb{H}^N) under the norm \|u\|=(\int_{\mathbb{H}^N}|\nabla_H u|^2+|u|^2)^{\frac{1}{2}}. Then we use this least energy solution to give a sharp estimate to the smallest positive constant in the Gagliardo-Nirenberg inequality on the Heisenberg group, 2 < p < 2+\frac{2}{N},

    \[\int_{\mathbb{H}^N}|u|^p\,d\xi \leq C\left(\int_{\mathbb{H}^N}|\nabla_H u|^2\,d\xi\right)^{\frac{Q(p-2)}{4}}\left(\int_{\mathbb{H}^N}|u|^2\,d\xi\right)^{\frac{2p-Q(p-2)}{4}}.\]

We also point out some extensions to the quasilinear sub-elliptic case.

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1007/978-3-0348-0516-2)
Extra info: (P085/2013-02-05)

Nonlinear Control Theory (10)

First integrals for problems of the calculus of variations on locally convex spaces (with D. Torres), E. J. Appl. Sci. 10(2008), 207-218.
(myid-9)

Quadratures of Pontryagin extremals for optimal control problems (with D. Torres), Control and Cybernetics 35(2006), 947-963.
(myid-27)

Symbolic computation of variational symmetries in optimal control (with P. Gouveia, D. Torres), Control and Cybernetics 35(2006), 832-849.
(myid-26)

Control-theoretic methods for design of algorithms of digital arithmetic (with A. Sarychev, A. Pereira, R. Rodrigues) J. Math. Sci. 120(2004), 995-1005.
(myid-21)

An algebraic approach to nonlinear control, PhD thesis, University of Aveiro, pp. 300, 2004.

(myid-22)

On computation of the logarithm of the Chen-Fliess series for nonlinear systems, in: Nonlinear and Adaptive Control, Zinober et al. (eds), Lect. Notes Control Inf. Sci. 281, 317-326, Springer-Verlag, 2003.
(myid-20)

An application of control-theoretic methods to digital arithmetic algorithms (with A. Sarychev, A. Pereira, R. Rodrigues), in: Proceedings of the 10th Mediterranean Conference on Control and Automation, Lisbon, Portugal, 2002.
(myid-17)

Coordinates of first kind and periodic time-variant feedback stabilization, in: Proceedings of the FJEM – Control Theory and Stabilization, Dijon, France, 2002.
(myid-18)

Time-variant feedback stabilization of a system of two rolling bodies (with A. Sarychev), in: Proceedings of the 5th Portuguese Conference on Automatic Control, Aveiro, Portugal, 2002.
(myid-16)

Controlos bang-bang e o fenómeno de Fuller em controlo óptimo, MSc dissertation, University of Aveiro, Portugal, 1998.
(myid-15)

Partial Differential Equations (31)

On the Schrödinger-Poisson system with a general indefinite nonlinearity (with L. Huang, J. Chen), Nonlinear Analysis: Real World Applications 28(2016), 1–19.

ABSTRACT: We study the existence and multiplicity of positive solutions of a class of Schrödinger-Poisson systems of the form

    \[\left\{ \begin{array}{ll} -\Delta u+u+l(x)\phi u = k(x)g(u) +\mu h(x)u \quad & \ \hbox{in} \ {\bkR}^3,\\ -\Delta \phi = l(x)u^2\quad & \ \hbox{in} \ {\bkR}^3, \end{array}\right.\]

where k\in C(\bkR^3) changes sign in \bkR^3, \lim_{|x|\rightarrow \infty} k(x)=k_\infty < 0, and the nonlinearity g behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that \mu > \mu_1 and near \mu_1, where \mu_1 is the first eigenvalue of -\Delta+id in H^1(\bkR^3) with weight function h, whose corresponding positive eigenfunction is denoted by e_1. An interesting phenomenon here is that we do not need the condition

    \[\int_{\mathbb{R}^3} k(x) e_1^p dx < 0,\]

which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. D.G. Costa, H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in \bkR^N, Calc. Var. Partial Differential Equations 13(2001) 159–189.)

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1016/j.nonrwa.2015.09.001)
Extra info: (P084/2016-04)

Sign changing solutions of a semilinear equation on Heisenberg group, (with J. Chen) accepted in J. Non. and Convex Anal. (2015).

ABSTRACT: This paper is concerned with the existence of multiple solutions to the semilinear equation

    \[\Delta_H\, u + |u|^{\frac{4}{Q-2}}u+\mu|\xi|^\alpha_H u=0\]

in a bounded domain of the Heisenberg group \mathbb{H}^N with Dirichlet boundary condition, where \alpha > 0 and |\xi|_H is a distance in \mathbb{H}^N. By using variational methods, we prove that this problem possesses at least one positive solution and one sign changing solution for some values of \alpha and \mu.

Download: Preprint version (RIA |  download) / Publisher version (not available yet)
Extra info: (P090/2015-06)

A reproducing kernel Hilbert discretization method for linear PDEs with nonlinear right-hand side, Lib. Math. 34:2(2014), 91-104.

ABSTRACT: In this paper, the main problem of study concerns to find a suitable discretization method to (numerically represent) the solution u\in\mathcal{H}, in a Hilbert space, of the general equation

(1)   \begin{equation*} L[u(x)]=f(x,u(x)),\quad x\in\Omega\subset\bkR^N, \end{equation*}

satisfying a given boundary condition (e.g. Dirichlet), where L is a linear differential (or integral) operator and f is a nonlinear function with enough regularity. The method Aveiro Discretization Method in Mathematics (ADMM), introduced by Saitoh et al. (2014; doi:10.1007/978-1-4939-1106-6 3), can deal with problem (1) when f is a function that do not depends on u. In fact, ADMM is a general method for solving by discretization, in a specific optimal sense and when a priori some data may be missing, a wide class of linear mathematical problems by using some key ideas of reproducing kernels and Tikhonov regularization theory. Here, we aim to extend the ADMM method to a more general situation where the nonlinearity may depend on u. Then, we apply the scheme to find the (optimal) discretization solution of the problem

    \[\vartheta\frac{\partial^2 u}{\partial y^2}+\alpha \frac{\partial^2 u}{\partial x^2}+\gamma u= u^3,\quad \mbox{ on } \bkR^2,\]

for arbitrary coefficient functions \vartheta, \alpha, \gamma.

Download: Preprint version (RIA |  download) / Publisher version (doi:10.14510/lm-ns.v34i2.1306)
Extra info: (P105/2014)

Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity (with L. Huang, J. Chen), J. Differential Equations 255(2013), 2463–2483.

ABSTRACT: We study the existence and multiplicity of positive solutions of a class of Schrödinger-Poisson systems of the form

    \[\left\{ \begin{array}{ll} -\Delta u+u+l(x)\phi u = k(x)g(u) +\mu h(x)u \quad & \ \hbox{in} \ {\bkR}^3,\\ -\Delta \phi = l(x)u^2\quad & \ \hbox{in} \ {\bkR}^3, \end{array}\right.\]

where k\in C(\bkR^3) changes sign in \bkR^3, \lim_{|x|\rightarrow \infty} k(x)=k_\infty < 0, and the nonlinearity g behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that \mu > \mu_1 and near \mu_1, where \mu_1 is the first eigenvalue of -\Delta+id in H^1(\bkR^3) with weight function h, whose corresponding positive eigenfunction is denoted by e_1. An interesting phenomenon here is that we do not need the condition \int_{\mathbb{R}^3} k(x) e_1^p dx < 0, which has been shown to be a sufficient condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity (see e.g. S. Alama, G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993) 439–475).

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1016/j.jde.2013.06.022)
Extra info: (P083/2013-10)

Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity (with L. Huang, J. Chen), J. Math. Anal. Appl. 408:1(2013), 55–69.

ABSTRACT: We consider the Schrödinger-Poisson system

    \[\left\{\begin{array}{ll} -\Delta u +u +l(x)\phi u = k(x)|u|^4u +\mu h(x)u \quad & \ \hbox{in} \ {\bkR}^3,\\ -\Delta \phi = l(x)u^2\quad & \ \hbox{in} \ {\bkR}^3, \end{array}\right.\]

where \mu is a positive constant and the nonlinear growth of |u|^4u reaches the Sobolev critical exponent, since 2^*=6 for three spatial dimensions. We prove the existence of (at least) a pair of fixed sign and a pair of sign-changing solutions in H^1({\bkR}^3)\times D^{1, 2}({\bkR}^3) under some suitable conditions on the non-negative functions l, k, h, but not requiring any symmetry property on them.

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1016/j.jmaa.2013.05.071)
Extra info: (P082/2013-06-04)

A positive solution of a Schrödinger-Poisson system with critical exponent (with L. Huang), Commun. Math. Anal. 15:1(2013), 29-43.

ABSTRACT: We use variational methods to study the existence of at least one positive solution of the following Schrödinger-Poisson system

    \[\left\{\begin{array}{ll} -\Delta u +u +l(x)\phi u = k(x)|u|^{{2^*}-2}u +\mu h(x)|u|^{q-2}u \quad & \ \hbox{in}\ {\bkR}^3,\\ -\Delta \phi = l(x)u^2\quad & \ \hbox{in} \ {\bkR}^3, \end{array}\right.\]

under some suitable conditions on the non-negative functions l, k, h and constant \mu > 0, where 2\leq q < 2^* (critical Sobolev exponent). Note that the nonlinearity involves a critical exponent, the Sobolev embedding H^1(\bkR^3)\hookrightarrow L^s(\bkR^3) (2\leq s\leq 6) is not compact. This will create additional difficulies in the proof of the Palais-Smale condition. We will transform the problem into a nonlocal elliptic equation in \mathbb{R}^3, where we also consider the limiting case q=2.

Download: Preprint version (RIA |  download) / Publisher version (Commun. Math. Anal.)
Extra info:> (P081/2013)

Existence of solutions of sub-elliptic equations on the Heisenberg group with critical growth and singularity (with J. Chen), Opuscula Math. 33:2(2013), 237-254.

ABSTRACT: We consider a class of sub-elliptic equations on the Heisenberg group \mathbb{H}^N with a Hardy type singularity and a critical nonlinear growth

    \[-\Delta_{H,p} u -\lambda\frac{|z|^p}{\rho^{2p}}|u|^{p-2}u = \frac{|z|^s}{\rho^{2s}}|u|^{p_\ast(s)-2}u\quad\mbox{ in }\mathbb{H}^N\backslash\{0\},\]

where \Delta_{H,p} is the p-sub-Laplacian with respect to a fixed bases for the corresponding Lie algebra of the Heisenberg group, (z,t)=(x,y,t)\in\mathbb{H}^N, \rho(\xi) is the distance of \xi\in\mathbb{H}^N to the origin given by the homogeneous distance

    \[d(\xi,\xi')=\left((|x-x'|^2+|y-y'|^2)^2+(t-t'-2(x,y')_{\mathbb{R}}-2(x',y)_{\mathbb{R}})^2\right)^{\frac{1}{4}},\]

Q=2N+2 is the homogeneous dimension, \mathcal{D}^{1,p}_0\left(\mathbb{H}^N\right) denotes the closure of C^\infty_0\left(\mathbb{H}^N\right) under the norm \|u\|=(\int_{\mathbb{H}^N}|\nabla_H u|^p\,d\xi)^{\frac{1}{p}} and p_\ast(s)=p(Q-s)(Q-p)^{-1} the critical exponent of embedding

    \[\mathcal{D}^{1,p}_0\left(\mathbb{H}^N\right) \hookrightarrow L^{p_\ast(s)}\left(\mathbb{H}^N\right, \frac{|z|^s}{\rho^{2s}}d\xi).\]

We prove the existence of energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252].

Download: Preprint version (RIA |  download) / Publisher version (doi:10.7494/OpMath.2013.33.2.237)
Extra info: (P080/2013) (free/open access)

A class of sub-elliptic equations on the Heisenberg group and related interpolation inequalities (with J. Chen) in Advances in Harmonic Analysis and Operator Theory, Operator Theory: Advances and Applications 229, 123–137, (2013).

ABSTRACT: We firstly prove the existence of least energy solutions to a class of sub-elliptic equations on the Heisenberg group \mathbb{H}^N of the form

    \[-\Delta_H u + u = |u|^{p-2}u\quad u\in S^{1,2}_0(\mathbb{H}^N),\]

where S^{1,2}_0(\mathbb{H}^N) is the closure of C^{\infty}_0(\mathbb{H}^N) under the norm \|u\|=(\int_{\mathbb{H}^N}|\nabla_H u|^2+|u|^2)^{\frac{1}{2}}. Then we use this least energy solution to give a sharp estimate to the smallest positive constant in the Gagliardo-Nirenberg inequality on the Heisenberg group, 2 < p < 2+\frac{2}{N},

    \[\int_{\mathbb{H}^N}|u|^p\,d\xi \leq C\left(\int_{\mathbb{H}^N}|\nabla_H u|^2\,d\xi\right)^{\frac{Q(p-2)}{4}}\left(\int_{\mathbb{H}^N}|u|^2\,d\xi\right)^{\frac{2p-Q(p-2)}{4}}.\]

We also point out some extensions to the quasilinear sub-elliptic case.

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1007/978-3-0348-0516-2)
Extra info: (P085/2013-02-05)

Four solutions of an elliptic problem with double singularity (with J. Chen, K. Murillo) Lib. Math. 32:2(2012), 169-192.

ABSTRACT: We consider the existence of nontrivial solutions u\in H_0^1(\Omega) of a Dirichlet problem with equation

    \[-\Delta u -\frac{\mu}{|x|^2}u = \lambda f(x)|u|^{q-2}u + \frac{|u|^{\frac{4-2s}{N-2}}u}{|x|^s},\quad x\in\Omega\backslash\{0\},\]

where 0\in\Omega\subset\mathbb{R}^N is a bounded domain with smooth boundary, f:\Omega \rightarrow \mathbb{R} is a real continuous and positive function, 0\leq s<2, 0\leq \mu < \bar{\mu} -4 (necessarily N> 6), and \frac{N+\sqrt{{\bar{\mu}-\mu}}}{\sqrt{\bar{\mu}} + \sqrt{\bar{\mu}-\mu}} < q < 2, with \bar{\mu} = (N-2)^2/4. Note that \frac{4-2s}{N-2} is the critical Sobolev-Hardy exponent minus 2. By variational methods we show that there are (at least) two positive solutions and (at least) one pair of sign-changing solutions in H_0^1(\Omega) for any \lambda\in (0,\Lambda^*), where \Lambda^* is a positive value suitably defined.

Download: Preprint version (RIA |  download) / Publisher version (Lib. Math.)
Extra info: (P066/2012-06-23)

Multiplicity result for a class of elliptic equations with singular term (with J. Chen, K. Murillo), Nonlinear Analysis TMA 75:15(2012), 5797–5814.

ABSTRACT: We consider the existence of nontrivial solutions of the equation

(1)   \begin{equation*} -\Delta u -{\frac{\lambda }{{|x|^{2}}}}u= |u|^{2^{\ast}-2}u + \mu|x|^{\alpha-2}u + f(x)|u|^{\gamma},\quad x\in \Omega\backslash\{0\},\:\:\: u\in H^1_0(\Omega), \end{equation*}

where 0\in\Omega is a smooth bounded domain in \mathbb{R}^N (N\geq 3). By variational methods and Nehari set techniques, we show that this problem, under some additional hypotheses on \lambda>0, \mu>0, \alpha>0, 0\leq \gamma<1 and f\in L^\infty(\Omega), has four nontrivial solutions in H^1_0(\Omega), and that least one of them is sign-changing. Classes of elliptic equations which include (1) have a lost of compactness phenomena, since the nonlinearity has a critical growth imposed by the critical exponent 2^{\ast } of the Sobolev embedding H^1_0(\Omega) into L^{2^{\ast }}(\Omega). This means that we could not use standard variational methods. On the other hand, due to the presence of the singular term \frac{\lambda }{{|x|^{2}}}, the problem has a strong singularity at 0\in \Omega.

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1016/j.na.2012.05.023)
Extra info: (P102/2012-06-23)

Existence of solutions of a class of nonlinear singular equations in Lorentz spaces (with L. Huang, K. Murillo), Advances in Harmonic Analysis and Operator Theory (the Stefan Samko anniversary volume, Integral Differential Operators and their Applications; Eds. Alexandre Almeida, Luis Castro, Frank-Olme Speck), pp.14, accepted in 2012.

Two nontrivial solutions of a class of elliptic equations with singular term (with J. Chen, K. Murillo), Dynamical Systems and Differential Equations, DCDS Supplement 2011. Proceedings of the 8th AIMS International Conference, Wei Feng et al., 272-281, Dresden (Germany), 2011. ISBN: 978-1-60133-007-9

ABSTRACT: We consider the existence of nontrivial solutions of the equation

    \[-\Delta u -{\frac{\lambda }{{|x|^{2}}}}u= |u|^{2^{\ast}-2}u + \mu|x|^{\alpha-2}u + f(x)|u|^{\gamma},\quad x\in \Omega\backslash\{0\},\:\:\: u\in H^1_0(\Omega),\]

where 0\in\Omega is a smooth bounded domain in \mathbb{R}^N (N\geq 3). By variational methods and Nehari set techniques, we show that this equation has at least two nontrivial solutions in H^1_0(\Omega), under some additional hypotheses on \lambda>0, \mu>0, \alpha>0, 0\leq\gamma<1 and f\in L^\infty(\Omega), which may be sign-changing. If f>0 then the solutions are positive.

Download: Preprint version (RIA |  download) / Publisher version (Discrete Contin. Dyn. Syst.)
Extra info: (P100/2011-09) (free/open access)

Existence of three nontrivial solutions for asymptotically p-linear noncoercive p-Laplacian equations (with N.S. Papageorgiou), Nonlinear Analysis TMA 74:16(2011), 5314–5326.
(myid-53)

Existence of stable standing waves and instability of standing waves to a class of quasilinear Schrödinger equations with potential (with J. Chen), Dynamics of PDEs 8:2(2011), 89-112.

ABSTRACT: We consider the following quasilinear Schrödinger equation with harmonic potential

(1)   \begin{equation*} i\varphi_t = -\triangle\varphi + |x|^2\varphi -|\varphi|^{p-1}\varphi - 2(\triangle|\varphi|^2)\varphi,\quad t\geq 0,\ x\in \mathbb{R}^N, \end{equation*}

where i^2 = 1, \varphi\equiv\varphi(t,x):\ \mathbb{R}_+\times\mathbb{R}^N\to \mathbb{C} is a complex-valued function and \triangle=\sum_{j=1}^N{{\partial^2}\over{\partial x_j^2}} is the standard Laplacian operator. We are concerned with stability and instability of standing wave solutions for (1). We will prove the existence of stable standing waves for 1 < p < 3 + {4\over N} and the existence of unstable standing waves for 3 + {4\over N}\leq p < 2\cdot 2^\ast - 1 (here and after, 2^\ast denotes the critical exponent, i.e. 2^\ast = {{2N}\over{N-2}} for N \geq 3 and 2^\ast = +\infty for N =1,\ 2). Our result indicates that the quasilinear term (\triangle|\varphi|^2)\varphi makes the standing wave more stable than their counterpart in the semilinear case, which is consistent with the physical phenomena and is in striking contrast with the classical semilinear Schrödinger equation.

Download: Preprint version (RIA |  download) / Publisher version (doi:10.4310/DPDE.2011.v8.n2.a2)
Extra info: (P071) (free/open access)

Sharp constant for a 2D anisotropic Sobolev inequality with critical nonlinearity (with J. Chen), Journal of Mathematical Analysis and Applications 367:2(2010), 685-692.

ASBTRACT: For the 2-dimensional anisotropic Sobolev inequality of the form

    \[\int_{\bkR^2} |u|^6dxdy \leq \alpha \bigg(\int_{\bkR^2} u_x^2dxdy\bigg)^2\int_{\mathbb{R}^2} |D_x^{-1}u_y|^2dxdy,\]

where D_x^{-1} is defined as

    \[D_x^{-1}h(x,y) = \int_{-\infty}^xh(s,y)ds,\]

it is proved that the sharp (smallest) positive constant \alpha is exactly given by 3(\int_{\mathbb{R}^2}\phi_x^2dxdy)^{-2}, where \phi is a minimal action solution of the equation

    \[(u_{xx} + |u|^4u)_x = D_x^{-1} u_{yy}.\]

Download: Preprint version (RIA | download) / Publisher version (doi:10.1016/j.jmaa.2010.02.020)
Extra info: (P073/2010-07-15)

On the existence of three nontrivial smooth solutions for nonlinear elliptic equations (with N.S. Papageorgiou, V. Staicu), pp.21, J. Nonlinear and Convex Analysis 11:1(2010), 115-136.

(myid-12)

On a class of quasilinear elliptic equations and related anisotropic Sobolev inequalities involving p-Laplace operator (with J. Chen), pp.15, submitted in 2011.

On a class of N-dimensional anisotropic Sobolev inequality and related results (with J. Chen), pp.31, preprint.

ABSTRACT: We determine the best (smallest) constant \alpha in the Anisotropic Sobolev inequality of the form

    \[\|u\|_p^p \leq \alpha\left\|u\right\|_2^{\frac{2(2N-1)+(3-2N)p}{2}}\,\|u_x\|_2^{\frac{N(p-2)}{2}} \,\prod_{k=1}^{N-1}\|D_x^{-1}\partial_{y_k}u\|_2^{\frac{p-2} {2}}\]

and the best (smallest) constant \beta in the inequality

    \[\|u\|_{p_*}^{p_*} \leq \beta \|u_x\|_2^{\frac{2N}{2N-3}}\, \prod_{k=1}^{N-1}\|D_x^{-1}\partial_{y_k}u\|_2^{\frac{2}{2N-3}},\]

where (x, y_1, \cdots, y_{N-1})\in \mathbb{R}^N with N\geq 3 and 2 < p < p_* = {\frac{2(2N-1)}{2N-3}}. These best constants are obtained by introducing a new method and using variational techniques. The method introduced here seems to have independent interest. We also use this method to find the best constant of the Gagliardo-Nirenberg interpolation inequality involving the m-Laplacian

    \[\|u\|_q^q \leq C_{Gm}\:\|\nabla u\|_m^{\frac{N(q-m)}{m}}\:\|u\|_m^{\frac{Nm-(N-m)q}{m}},\qquad u\in W^{1,m}(\mathbb{R}^N),\]

where 1 < m < q < m^\ast and m^\ast = Nm/(N-m) for 1 < m < N and m^\ast = \infty for m \geq N.

Extra info: (P074)

Positive solutions for elliptic problems with critical nonlinearity and combined singularity (with J. Chen), Mathematica Bohemica, 135:4(2010), 413-422.

ABSTRACT: We study the existence of multiple positive weak solutions of the equation

    \[\left\{\begin{array}{rll} -\Delta u - {\lambda\over {|x|^2}}u &= u^{2^\ast-1} + \mu u^{-q} & \hbox{in}\quad \Omega\backslash\{0\},\\ u(x) > 0 & \hbox{in}\quad \Omega\backslash\{0\},\quad\quad u(x)= 0 & \hbox{on}\quad \partial\Omega,\end{array}\right.\eqno{(P_{\lambda,\mu})}\]

where 0\in \Omega and \Omega\subset\mathbb{R}^N(N\geq 3) is a bounded domain with smooth boundary, 2^{\ast} =2N/(N-2) is the critical Sobolev exponent, 0 < \lambda < \Lambda=(N-2)^2/4 and 0 < q < 1. We use variational methods to prove that for suitable \mu, the problem has at least two positive weak solutions.

Download: Preprint version (RIA |  download) / Publisher version (Math. Bohemica)
Extra info: (P072) (free/open access)

The convergence analysis of the decomposition method for the (1+1)-parabolic problem in nonuniform media (with M.M. Rodrigues), Acta Applicanda Mathematicae, 2010, Volume 112(3) 299-308.

(myid-81)

Twin positive solutions for singular nonlinear elliptic equations (with J. Chen, N.S. Papageorgiou), Topological Methods in Nonlinear Analysis 35:1(2010), 187-201.

ABSTRACT: For a bounded domain \Omega\subseteq\bkR^N with a C^2-boundary, we prove the existence of an ordered pair of smooth positive strong solutions for the nonlinear Dirichlet problem

    \[-\Delta_p\, u(x) = \beta(x)u(x)^{-\eta}+f(x,u(x)) \mbox{ a.e. on } \Omega \quad\mbox{with } u\in W^{1,p}_0(\Omega),\]

which exhibits the combined effects of a singular term (\eta\geq 0) and a (p-1)-linear term f(x,u) near +\infty, by using a combination of variational methods, with upper-lower solutions and with suitable truncation techniques.

Download: Preprint version (RIA | download) / Publisher version (TMNA)
Remark: The paper follows Papageorgiou’s notation
Extra info: (P067/2010-03-01)

Existence and multiplicity of solutions for the noncoercive Neumann p-Laplacian (with N.S. Papageorgiou), Electronic Journal of Differential Equations, Conference 18 (2010), pp. 57-66,
(special issue of the Conference “Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems 2007”, in 2009).
(myid-38)

On approximate solutions to the wavefront speed of reaction-diffusion-convection problem in nonuniform media (with M.M. Rodrigues), Asymptotic Analysis 66(2010) 51–59.
(myid-10)

Four solutions of an inhomogeneous elliptic equation with critical exponent and singular term (with J. Chen), Nonlinear Analysis TMA 71:10(2009), 4739-4750.

ABSTRACT: In this paper, we prove the existence of four nontrivial solutions of

    \[-\Delta u - {\lambda\over {|x|^2}}u = |u|^{2^{\ast}-2}u + \mu|x|^{\alpha-2}u + f(x),\quad\quad x\in \Omega\backslash\{0\},\]

 and show that at least one of them is sign changing. Our results extend some previous works on the literature, as Tarantello(1993), Kang-Deng(2005) and Hirano-Shioji(2007).

Download: Preprint version (RIA |  download) / Publisher version (doi:10.1016/j.na.2009.03.048)
Extra info: (P057/2009-11-15)

On nonlinear parametric problems for p-Laplacian-like operators (with N.S. Papageorgiou), Rev. R. Acad. Cien. Serie A. Mat. 103:1(2009), 177–200.
(myid-60)

Exact and approximate solutions of reaction-diffusion-convection equations (with M.M. Rodrigues), AIP-Proceedings of International Conference on Boundary Value Problems-BVP 2008, A. Cabada, E. Liz, J.J. Nieto, 10 pp, Santiago de Compostela, 2009.
(myid-11)

Pairs of positive solutions for p-Laplacian equations with sublinear and superlinear nonlinearities which do not satisfy the AR-condition (with N.S. Papageorgiou), Nonlinear Analysis TMA 70(2009), 3854-3863.
(myid-37)

A multiplicity theorem for a variable exponent Dirichlet problem (with N.S. Papageorgiou), Glasgow Math. Journal 50(2008), 1-15.
(myid-29)

A multiplicity theorem for hemivariational inequalities with a p-Laplacian-like differential operator (with N.S. Papageorgiou, V. Staicu), Nonlinear Analysis 69(2008), 1150-1163.
(myid-4)

Multiplicity theorems for superlinear elliptic problems (with N.S. Papageorgiou, V. Staicu), Calculus of Variations and Partial Differential Equations 33:2(2008), 199-230.

ABSTRACT: Let \Omega\subseteq\bkR^N be a bounded domain with a C^2-boundary \partial \Omega. In this paper we study second order elliptic equations of the form

    \[\left\{ \begin{array}{ll} -\mbox{div} \left(|Du|^{p-2}Du(x)\right)=f(x,u(x)) & \mbox{ a.e. on }\Omega,\\ \left. u\right|_{\partial \Omega}=0, & 1<p<\infty. \end{array} \right.\]

driven by the Laplacian and p-Laplacian differential operators and a nonlinearity which is (p-)superlinear (it satisfies the Ambrosetti-Rabinowitz condition). For the p-Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. For the semi linear problems using in addition Morse theory, we obtain six nontrivial solutions. We prove seven such multiplicity results. The first five concern problems driven by the p-Laplacian, while the last two deal with the particular case p=2 (semilinear problems). In all these theorems we also provide precise sign information about the solutions.

Download: Preprint version (RIA | download) / Publisher version (doi:10.1007/s00526-008-0172-7)
Remark: The paper follows Papageorgiou’s notation
Extra info: (P006/2008-03-08)

The speed of reaction-diffusion-convection wavefronts in nonuniform media (with M.M. Rodrigues), American Institute of Physics 936(2007), 666-670.
(myid-28)