Advanced Nonlinear Studies

Harrabi, Abdellaziz; Rebhi, Salem

doi: 10.1515/ans-2010-0401pmid: N/A

AbstractWe study here finite Morse index solutions of -∆u = f(u) on the entire space or half space and their application to smooth bounded domain problems when the growth of the non-linearity is faster than the usual Sobolev critical exponent.

Harrabi, Abdellaziz; Rebhi, Salem

doi: 10.1515/ans-2010-0401pmid: N/A

Abstract We study here finite Morse index solutions of -∆u = f(u) on the entire space or half space and their application to smooth bounded domain problems when the growth of the non-linearity is faster than the usual Sobolev critical exponent.

Goncalves, J. V. A.; Zhou, Jiazheng

doi: 10.1515/ans-2010-0402pmid: N/A

AbstractWe deal with existence of large solutions of ∆pu = a(x)f(u)+b(x)g(u) in ℝN. It is shown that if a, b, f, g are non-negative real valued functions with a, b ∈ C(ℝN), f, g ∈ C([0,∞)) and f + g ≥ h where h is a continuous, non-negative, non- decreasing function satisfying the Keller-Osserman condition then the equation above admits a large solution if the equation -∆pv = a(x) + b(x) in ℝN has a positive upper solution decaying to zero at infinity. No monotonicity condition is required from either f or g. Our proof is based on the method of lower and upper-solutions. We extend recent results by A. V. Lair and A. Mohammed.

Omrane, Ines Ben; Jleli, Mohamed; Samet, Bessem

doi: 10.1515/ans-2010-0403pmid: N/A

AbstractIn this paper, we give a weighted version of regularity of solutions of the Wente problem associated to the modified Helmholtz operator -Δ + αI, where α is a positive constant.

Montoro, Luigi; Sciunzi, Berardino; Squassina, Marco

doi: 10.1515/ans-2010-0404pmid: N/A

Abstract We study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic equations is also investigated. In particular, if the domain is a ball, the elements of the ω limit set are nonnegative radially symmetric solutions of the stationary problem.

Montoro, Luigi; Sciunzi, Berardino; Squassina, Marco

doi: 10.1515/ans-2010-0404pmid: N/A

AbstractWe study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic equations is also investigated. In particular, if the domain is a ball, the elements of the ω limit set are nonnegative radially symmetric solutions of the stationary problem.

Gasiński, Leszek; Papageorgiou, Nikolaos S.

doi: 10.1515/ans-2010-0405pmid: N/A

AbstractWe consider semilinear periodic ordinary differential equations with the nonlinearity exhibiting double resonance at infinity in the spectral interval [λk, λk+1], k ≥ 1. Using minimax methods based on the critical point theory together with Morse theory, we show that the problem has at least four nontrivial solutions.

Gasiński, Leszek; Papageorgiou, Nikolaos S.

doi: 10.1515/ans-2010-0405pmid: N/A

Abstract We consider semilinear periodic ordinary differential equations with the nonlinearity exhibiting double resonance at infinity in the spectral interval (λ k , λ k+1 ), k ≥ 1. Using minimax methods based on the critical point theory together with Morse theory, we show that the problem has at least four nontrivial solutions.

Mammeri, Youcef

doi: 10.1515/ans-2010-0406pmid: N/A

Abstract We are interested in dispersive properties of the Boussinesq system for small initial data. We prove that, for a suficiently high order nonlinearity, the solution of the Boussinesq system exists for all time and tends to zero as the time tends to infinity.

Mammeri, Youcef

doi: 10.1515/ans-2010-0406pmid: N/A

AbstractWe are interested in dispersive properties of the Boussinesq system for small initial data. We prove that, for a suficiently high order nonlinearity, the solution of the Boussinesq system exists for all time and tends to zero as the time tends to infinity.