Radial positive solutions for Neumann problems without growth restrictions

2016 ◽  
Vol 62 (6) ◽  
pp. 848-861 ◽  
Author(s):  
Ruyun Ma ◽  
Hongliang Gao ◽  
Tianlan Chen
Keyword(s):  
Positive Solutions ◽  
Neumann Problems ◽  
Growth Restrictions

Multiple positive solutions of some quasilinear neumann problems

2000 ◽  
Vol 74 (3-4) ◽  
pp. 363-391 ◽  
Author(s):  
Stanislav I. Pohozaev ◽  
Laurent Véron
Keyword(s):  
Positive Solutions ◽  
Multiple Positive Solutions ◽  
Neumann Problems

scholarly journals Unbounded principal eigenfunctions and the logistic equation on RN

2003 ◽  
Vol 67 (3) ◽  
pp. 413-427 ◽  
Author(s):  
Wei Dong ◽  
Yihong Du
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Growth Conditions ◽  
Logistic Equation ◽  
Exponential Rate ◽  
Unique Positive Solution ◽  
Unbounded Coefficients ◽  
Growth Restrictions

We consider the logistic equation − Δu = a (x) u − b (x) up on all of RN with possibly unbounded coefficients near infinity. We show that under suitable growth conditions of the coefficients, the behaviour of the positive solutions of the logistic equation can be largely determined. We also show that certain linear eigenvalue problems on all of RN have principal eigenfunctions that become unbounded near infinity at an exponential rate. Using these results, we finally show that the logistic equation has a unique positive solution under suitable growth restrictions for its coefficients.

scholarly journals A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth

2019 ◽  
Vol 150 (1) ◽  
pp. 73-102 ◽  
Author(s):  
Alberto Boscaggin ◽  
Francesca Colasuonno ◽  
Benedetta Noris
Keyword(s):  
Positive Solutions ◽  
Critical Power ◽  
A Priori ◽  
Oscillatory Behaviour ◽  
Radial Solutions ◽  
A Priori Bounds ◽  
Shooting Technique ◽  
Neumann Problems ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

AbstractLet 1 < p < +∞ and let Ω ⊂ ℝN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type $-\Delta _pu = f(u),\quad u > 0\,{\rm in }\,\Omega ,\quad \partial _\nu u = 0\,{\rm on }\,\partial \Omega .$We suppose that f(0) = f(1) = 0 and that f is negative between the two zeros and positive after. In case Ω is a ball, we also require that f grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focussing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behaviour (around 1) of non-constant radial solutions.

scholarly journals Existence and nonexistence of positive solutions for parametric Neumann problems with $p$-Laplacian

2014 ◽  
Vol 66 (1) ◽  
pp. 137-153 ◽  
Author(s):  
Dumitru Motreanu ◽  
Viorica V. Motreanu ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Positive Solutions ◽  
Neumann Problems ◽  
Existence And Nonexistence

scholarly journals Positive solutions of Neumann problems with singularities

2008 ◽  
Vol 337 (2) ◽  
pp. 1267-1272 ◽  
Author(s):  
Jifeng Chu ◽  
Yigang Sun ◽  
Hao Chen
Keyword(s):  
Positive Solutions ◽  
Neumann Problems
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