scholarly journals Exact Multiplicity of Sign-Changing Solutions for a Class of Second-Order Dirichlet Boundary Value Problem with Weight Function

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yulian An
Keyword(s):  
Boundary Value Problem ◽  
Weight Function ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Constant Sign ◽  
Dirichlet Boundary Value Problem ◽  
Multiplicity Results ◽  
Sign Changing Solutions ◽  
Exact Multiplicity

Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problemsu″+a(t)f(u)=0,t∈(0,1),u(0)=0, andu(1)=0, wheref∈C(ℝ,ℝ)satisfiesf(0)=0and the limitsf∞=lim|s|→∞(f(s)/s),f0=lim|s|→0(f(s)/s)∈{0,∞}. Weight functiona(t)∈C1[0,1]satisfiesa(t)>0on[0,1].

New exact multiplicity results with an application to a population model

2001 ◽  
Vol 131 (5) ◽  
pp. 1167-1182 ◽  
Author(s):  
Philip Korman ◽  
Junping Shi
Keyword(s):  
Boundary Value Problem ◽  
Population Model ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Value Problem ◽  
Multiplicity Results ◽  
Linearized Problem ◽  
Positivity Of Solutions ◽  
Image Position ◽  
Exact Multiplicity ◽  
Crucial Part

We obtain some new exact multiplicity results for the Dirichlet boundary-value problem on a unit ball Bn in Rn. We consider several classes of nonlinearities f(u), including both positive and sign-changing cases. A crucial part of the proof is to establish positivity of solutions for the corresponding linearized problem. As an application we obtain exact multiplicity results for the Holling-Tanner population model.

scholarly journals SOME MULTIPLICITY RESULTS TO THE EXISTENCE OF THREE SOLUTIONS FOR A DIRICHLET BOUNDARY VALUE PROBLEM INVOLVING THE P-LAPLACIAN

2011 ◽  
Vol 16 (3) ◽  
pp. 390-400 ◽  
Author(s):  
Shapour Heidarkhani ◽  
Ghasem Alizadeh Afrouzi
Keyword(s):  
Boundary Value Problem ◽  
Critical Points ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Main Tool ◽  
Dirichlet Boundary Value Problem ◽  
Positive Real ◽  
Three Solutions ◽  
Multiplicity Results ◽  
Uniformly Bounded

In this paper we prove the existence of two intervals of positive real parameters λ for a Dirichlet boundary value problem involving the p-Laplacian which admit three weak solutions, whose norms are uniformly bounded with respect to λ belonging to one of the two intervals. Our main tool is a three critical points theorem due to G. Bonanno [A critical points theorem and nonlinear differential problems, J. Global Optim., 28:249–258, 2004].

scholarly journals Existence theory for single and multiple solutions to semipositone discrete Dirichlet boundary value problems with singular dependent nonlinearities

2003 ◽  
Vol 16 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Daqing Jiang ◽  
Lili Zhang ◽  
Donal O'Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Existence Theory ◽  
Dirichlet Boundary Value Problem

In this paper we establish the existence of single and multiple solutions to the semipositone discrete Dirichlet boundary value problem {Δ2y(i−1)+μf(i,y(i))=0,            i∈{1,2,…,T}y(0)=y(T+1)=0, where μ>0 is a constant and our nonlinear term f(i,u) may be singular at u=0.

scholarly journals Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1907
Author(s):  
Valery Karachik
Keyword(s):  
Boundary Value Problem ◽  
Unit Ball ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Dirichlet Boundary Value Problem ◽  
Polyharmonic Equation ◽  
Harmonic Components

In the previous author’s works, a representation of the solution of the Dirichlet boundary value problem for the biharmonic equation in terms of Green’s function is found, and then it is shown that this representation for a ball can be written in the form of the well-known Almansi formula with explicitly defined harmonic components. In this paper, this idea is extended to the Dirichlet boundary value problem for the polyharmonic equation, but without invoking the Green’s function. It turned out to find an explicit representation of the harmonic components of the m-harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of m solutions to the Dirichlet boundary value problems for the Laplace equation in the unit ball. Then, using this representation, an explicit formula for the harmonic components of the solution to the Neumann boundary value problem for the polyharmonic equation in the unit ball is obtained. Examples are given that illustrate all stages of constructing solutions to the problems under consideration.

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