Quasilinear Schrödinger equations involving concave and convex nonlinearities
João Marcos do Ó Uberlandio Severo
Communications on Pure & Applied Analysis 2009, 8(2): 621-644 doi: 10.3934/cpaa.2009.8.621
In this paper the mountain--pass theorem and the Ekeland variational principle in a suitable Orlicz space are employed to establish the existence of positive standing wave solutions for a quasilinear Schrödinger equation involving a combination of concave and convex terms. The second order nonlinearity considered in this paper corresponds to the superfluid equation in plasma physics.
keywords: Schrödinger equations Orlicz spaces nonlinearity with concave and convex terms standing wave solutions variational methods
Solutions for singular quasilinear Schrödinger equations with one parameter
João Marcos do Ó Abbas Moameni
Communications on Pure & Applied Analysis 2010, 9(4): 1011-1023 doi: 10.3934/cpaa.2010.9.1011
We establish the existence of positive bound state solutions for the singular quasilinear Schrödinger equation

$i\frac{\partial \psi}{\partial t}=- \Delta \psi+\psi + \bar{\omega} (|\psi |^2)\psi- \lambda \rho(|\psi|^2)\psi-\kappa\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi, x \in \Omega,$

where $\bar{\omega} (\tau^2) \tau \rightarrow +\infty$ as $\tau \rightarrow 0$ and, $\lambda>0$ is a parameter and $\Omega$ is a ball in $\mathcal{R}^N$. This problem is studied in connection with the following quasilinear eigenvalue problem

$-\Delta \Psi-\kappa\Delta \rho(|\Psi|^2)\rho'(|\Psi|^2)\Psi =\lambda_1 \rho(|\Psi|^2)\Psi, x \in \Omega,$

Indeed, we establish the existence of solutions for the above Schrödinger equation when $\lambda$ belongs to a certain neighborhood of the first eigenvalue $\lambda_1$ of the above eigenvalue problem. The main feature of this paper is that the nonlinearity $ \bar{\omega} ( |\psi |^2)\psi$ is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter $\lambda$ combined with the nonlinear nonhomogeneous term $-\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi$. The proofs are based on various techniques related to variational methods and implicit function theorem.

keywords: variational methods. solitary waves Schrödinger equations
Positive radial solutions for some quasilinear elliptic systems in exterior domains
João Marcos do Ó Sebastián Lorca Justino Sánchez Pedro Ubilla
Communications on Pure & Applied Analysis 2006, 5(3): 571-581 doi: 10.3934/cpaa.2006.5.571
We use fixed-point theorem of cone expansion/compression type to prove the existence of positive radial solutions for the following class of quasilinear elliptic systems in exterior domains

$-\Delta_p u = k_1(|x| )f(u,v),$ for $|x| > 1$ and $x \in \mathbb R^N, $

$-\Delta_p v = k_2(|x|)g(u,v),$ for $|x| > 1 $ and $x \in \mathbb R^N, $

$u(x) = v(x) =0,$ for $|x| =1, $

$u(x), v(x) \rightarrow 0 $ as $|x| \rightarrow +\infty,$

where $1 < p < N $ and $\Delta_p u=$ div $(|\nabla u|^{p-2}\nabla u )$ is the p-Laplacian operator. We consider nonlinearities that are either superlinear or sublinear.

keywords: Elliptic systems exterior domains positive radial solutions p-Laplacian.
Elliptic equations and systems with critical Trudinger-Moser nonlinearities
Djairo G. De Figueiredo João Marcos do Ó Bernhard Ruf
Discrete & Continuous Dynamical Systems - A 2011, 30(2): 455-476 doi: 10.3934/dcds.2011.30.455
In this article we give first a survey on recent results on some Trudinger-Moser type inequalities, and their importance in the study of nonlinear elliptic equations with nonlinearities which have critical growth in the sense of Trudinger-Moser. Furthermore, recent results concerning systems of such equations will be discussed.
keywords: variational methods Trudinger-Moser inequality critical growth.
Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations
Daniele Cassani João Marcos do Ó Abbas Moameni
Communications on Pure & Applied Analysis 2010, 9(2): 281-306 doi: 10.3934/cpaa.2010.9.281
In this paper we prove the existence and qualitative properties of positive bound state solutions for a class of quasilinear Schrödinger equations in dimension $N\ge 3$: we investigate the case of unbounded potentials which can not be covered in a standard function space setting. This leads us to use a nonstandard penalization technique, in a borderline Orlicz space framework, which enable us to build up a one parameter family of classical solutions which have finite energy and exhibit, as the parameter goes to zero, a concentrating behavior around some point which is localized.
keywords: local mountain pass Quasilinear Schrödinger equation Orlicz spaces solitary waves variational methods. concentration phenomena

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