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.

Positive Solutions for Systems of Quasilinear Equations with Non-homogeneous Operators and Weights

2020 ◽  
Vol 20 (2) ◽  
pp. 293-310
Author(s):  
Marta García-Huidobro ◽  
Raúl Manasevich ◽  
Satoshi Tanaka
Keyword(s):  
Positive Solutions ◽  
Differential Operators ◽  
Blow Up ◽  
A Priori ◽  
Degree Theory ◽  
Radial Solutions ◽  
A Priori Bounds ◽  
Existence Of A Solution ◽  
Weakly Coupled System ◽  
Positive Radial Solutions

AbstractIn this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

Existence and multiplicity of positive solutions for the nonlinear Schrödinger–Poisson equations

2013 ◽  
Vol 143 (4) ◽  
pp. 745-764 ◽  
Author(s):  
Ching-yu Chen ◽  
Yueh-cheng Kuo ◽  
Tsung-fang Wu
Keyword(s):  
Positive Solutions ◽  
Nonlinear Schrödinger ◽  
Poisson Equations ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

We study the existence and multiplicity of positive solutions for the following nonlinear Schrödinger–Poisson equations: where 2 < p ≤ 3 or 4 ≤ p < 6, λ > 0 and Q ∈ C(ℝ3). We show that the number of positive solutions is dependent on the profile of Q(x).

Existence and Multiplicity of Positive Solutions for Singular Semipositone p-Laplacian Equations

2006 ◽  
Vol 58 (3) ◽  
pp. 449-475 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Daomin Cao ◽  
Haishen Lü ◽  
Donal O'Regan
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Solution Method ◽  
Degree Theory ◽  
Lower Solution ◽  
Existence And Multiplicity ◽  
Lower Solution Method ◽  
Multiplicity Of Positive Solutions ◽  
Image Position ◽  
Laplacian Equations

AbstractPositive solutions are obtained for the boundary value problemHere f (t, u) ≥ –M, (M is a positive constant) for (t, u) ∈ [0, 1]×(0, ∞). We will show the existence of two positive solutions by using degree theory together with the upper–lower solution method.

scholarly journals Existence and Multiplicity of Positive Solutions for a System of Fourth-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Shoucheng Yu ◽  
Zhilin Yang
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Index Theory ◽  
Existence And Multiplicity ◽  
Fixed Point Index Theory ◽  
Multiplicity Of Positive Solutions

We study the existence and multiplicity of positive solutions for the system of fourth-order boundary value problems x(4)=ft,x,x′,-x′′,-x′′′,y,y′,-y′′,-y′′′,  y(4)=gt,x,x′,-x′′,-x′′′,y,y′,-y′′,-y′′′,  x(0)=x′(1)=x′′(0)=x′′′(1)=0, and y(0)=y′(1)=y′′(0)=y′′′(1)=0, where f,g∈C([0,1]×R+8,R+)  (R+:=[0,∞)). We use fixed point index theory to establish our main results based on a priori estimates achieved by utilizing some integral identities and inequalities and R+2-monotone matrices.

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