scholarly journals Increasing radial solutions for Neumann problems without growth restrictions

2012 ◽  
Vol 29 (4) ◽  
pp. 573-588 ◽  
Author(s):  
Denis Bonheure ◽  
Benedetta Noris ◽  
Tobias Weth
Keyword(s):  
Radial Solutions ◽  
Neumann Problems ◽  
Growth Restrictions

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 Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities

2010 ◽  
Vol 28 (2) ◽  
pp. 637-648 ◽  
Author(s):  
Cristian Bereanu ◽  
◽  
Petru Jebelean ◽  
Jean Mawhin ◽  
◽  
...  
Keyword(s):  
Radial Solutions ◽  
Neumann Problems

scholarly journals Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions

2018 ◽  
Vol 24 (4) ◽  
pp. 1625-1644 ◽  
Author(s):  
Alberto Boscaggin ◽  
Francesca Colasuonno ◽  
Benedetta Noris
Keyword(s):  
Shooting Method ◽  
Radial Solution ◽  
Growth Conditions ◽  
Oscillatory Behavior ◽  
Multiple Positive Solutions ◽  
Radial Solutions ◽  
Neumann Problems ◽  
Existence And Multiplicity ◽  
Constant Solution ◽  
Neumann Laplacian

For 1 < p < ∞, we consider the following problem      −Δpu = f(u),   u > 0 in Ω,   ∂νu = 0 on ∂Ω, where Ω ⊂ ℝN is either a ball or an annulus. The nonlinearity f is possibly supercritical in the sense of Sobolev embeddings; in particular our assumptions allow to include the prototype nonlinearity f(s) = −sp−1 + sq−1 for every q > p. We use the shooting method to get existence and multiplicity of non-constant radial solutions. With the same technique, we also detect the oscillatory behavior of the solutions around the constant solution u ≡ 1. In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris and T. Weth, Ann. Inst. Henri Poincaré Anal. Non Linéaire 29 (2012) 573−588], that is to say, if p = 2 and f′ (1) > λradk + 1, with λradk + 1 the (k + 1)-th radial eigenvalue of the Neumann Laplacian, there exists a radial solution of the problem having exactly k intersections with u ≡ 1, for a large class of nonlinearities.

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