**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 in the Anisotropic Sobolev inequality of the form

and the best (smallest) constant in the inequality

where with and . 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 -Laplacian

where and for and for .

**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

where changes sign in , , and the nonlinearity behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that and near , where is the first eigenvalue of in with weight function , whose corresponding positive eigenfunction is denoted by . An interesting phenomenon here is that we do not need the condition

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 , Calc. Var. Partial Differential Equations 13(2001) 159–189.)

**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

in a bounded domain of the Heisenberg group with Dirichlet boundary condition, where and is a distance in . By using variational methods, we prove that this problem possesses at least one positive solution and one sign changing solution for some values of and .

**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 in a Hilbert space, of the general equation

(1)

satisfying a given boundary condition (e.g. Dirichlet), where is a linear differential (or integral) operator and 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 is a function that do not depends on . 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 . Then, we apply the scheme to find the (optimal) discretization solution of the problem

for arbitrary coefficient functions .

**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

where changes sign in , , and the nonlinearity behaves like a power at zero and at infinity. We mainly prove the existence of at least two positive solutions in the case that and near , where is the first eigenvalue of in with weight function , whose corresponding positive eigenfunction is denoted by . An interesting phenomenon here is that we do not need the condition , 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).

**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

where is a positive constant and the nonlinear growth of reaches the Sobolev critical exponent, since for three spatial dimensions. We prove the existence of (at least) a pair of fixed sign and a pair of sign-changing solutions in under some suitable conditions on the non-negative functions , but not requiring any symmetry property on them.

**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

under some suitable conditions on the non-negative functions and constant , where (critical Sobolev exponent). Note that the nonlinearity involves a critical exponent, the Sobolev embedding 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 , where we also consider the limiting case .

**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 with a Hardy type singularity and a critical nonlinear growth

where is the -sub-Laplacian with respect to a fixed bases for the corresponding Lie algebra of the Heisenberg group, , is the distance of to the origin given by the homogeneous distance

is the homogeneous dimension, denotes the closure of under the norm and the critical exponent of embedding

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].

**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 of the form

where is the closure of under the norm . 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, ,

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

**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 of a Dirichlet problem with equation

where is a bounded domain with smooth boundary, is a real continuous and positive function, , (necessarily ), and , with . Note that 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 for any , where is a positive value suitably defined.

**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)

where is a smooth bounded domain in (). By variational methods and Nehari set techniques, we show that this problem, under some additional hypotheses on , , , and , has four nontrivial solutions in , 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 of the Sobolev embedding into . This means that we could not use standard variational methods. On the other hand, due to the presence of the singular term , the problem has a strong singularity at .

*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

where is a smooth bounded domain in (). By variational methods and Nehari set techniques, we show that this equation has at least two nontrivial solutions in , under some additional hypotheses on , , , and , which may be sign-changing. If then the solutions are positive.

**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)

where , is a complex-valued function and 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 and the existence of unstable standing waves for (here and after, denotes the critical exponent, i.e. for and for ). Our result indicates that the quasilinear term 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.

**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

where is defined as

it is proved that the sharp (smallest) positive constant is exactly given by , where is a minimal action solution of the equation

**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)*

**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

where and () is a bounded domain with smooth boundary, is the critical Sobolev exponent, and . We use variational methods to prove that for suitable , the problem has at least two positive weak solutions.

**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 with a -boundary, we prove the existence of an ordered pair of smooth positive strong solutions for the nonlinear Dirichlet problem

which exhibits the combined effects of a singular term () and a -linear term near , by using a combination of variational methods, with upper-lower solutions and with suitable truncation techniques.

**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

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 be a bounded domain with a -boundary . In this paper we study second order elliptic equations of the form

driven by the Laplacian and -Laplacian differential operators and a nonlinearity which is (-)superlinear (it satisfies the Ambrosetti-Rabinowitz condition). For the -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 -Laplacian, while the last two deal with the particular case (semilinear problems). In all these theorems we also provide precise sign information about the solutions.

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

*(myid-28)*