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 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].

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 OF INFINITELY MANY SOLUTIONS FOR SUBLINEAR ELLIPTIC PROBLEMS

2012 ◽  
Vol 54 (3) ◽  
pp. 535-545
Author(s):  
X. ZHONG ◽  
W. ZOU
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Elliptic Problems ◽  
Infinitely Many Solutions ◽  
Dirichlet Boundary Value Problem ◽  
Image Position

AbstractWe study the following nonlinear Dirichlet boundary value problem: where Ω is a bounded domain in ℝN(N ≥ 2) with a smooth boundary ∂Ω and g ∈ C(Ω × ℝ) is a function satisfying $\displaystyle \underset{|t|\rightarrow 0}{\lim}\frac{g(x, t)}{t}= \infty$ for all x ∈ Ω. Under appropriate assumptions, we prove the existence of infinitely many solutions when g(x, t) is not odd in t.

scholarly journals Variational Method to p-Laplacian Fractional Dirichlet Problem with Instantaneous and Noninstantaneous Impulses

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yiru Chen ◽  
Haibo Gu ◽  
Lina Ma
Keyword(s):  
Dirichlet Problem ◽  
Variational Method ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Major Result ◽  
Dirichlet Boundary Value Problem ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

In this paper, a research has been done about the existence of solutions to the Dirichlet boundary value problem for p-Laplacian fractional differential equations which include instantaneous and noninstantaneous impulses. Based on the critical point principle and variational method, we provide the equivalence between the classical and weak solutions of the problem, and the existence results of classical solution for our equations are established. Finally, an example is given to illustrate the major result.

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