Uniqueness and exact multiplicity results for two classes of semilinear problems

1998 ◽  
Vol 31 (7) ◽  
pp. 849-865 ◽  
Author(s):  
Philip Korman
Keyword(s):  
Multiplicity Results ◽  
Semilinear Problems ◽  
Exact Multiplicity

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 Perturbation of Global Solution Curves for Semilinear Problems

2003 ◽  
Vol 3 (2) ◽  
Author(s):  
Philip Korman ◽  
Yi Li ◽  
Tiancheng Ouyang
Keyword(s):  
Global Solution ◽  
Positive Solutions ◽  
Turning Points ◽  
Dirichlet Problems ◽  
Indirect Approach ◽  
Smooth Curves ◽  
Semilinear Problems ◽  
Multiplicity Of Positive Solutions ◽  
Exact Multiplicity

AbstractWe revisit the question of exact multiplicity of positive solutions for a class of Dirichlet problems for cubic-like nonlinearities, which we studied in [6]. Instead of computing the direction of bifurcation as we did in [6], we use an indirect approach, and study the evolution of turning points. We give conditions under which the critical (turning) points continue on smooth curves, which allows us to reduce the problem to the easier case of f (0) = 0. We show that the smallest root of f (u) does not have to be restricted.

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].

Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems

2004 ◽  
Vol 134 (1) ◽  
pp. 179-190 ◽  
Author(s):  
Philip Korman
Keyword(s):  
Fourth Order ◽  
Bifurcation Theory ◽  
Elastic Beam ◽  
Uniqueness Results ◽  
Semilinear Problems ◽  
Linearized Problem ◽  
Nonlinear Force ◽  
Exact Multiplicity Of Solutions ◽  
Exact Multiplicity ◽  
Crucial Part

Using techniques of bifurcation theory, we give exact multiplicity and uniqueness results for the fourth-order Dirichlet problem, which describes deflection of an elastic beam, subjected to a nonlinear force, and clamped at the end points. The crucial part of this approach was to show positivity of non-trivial solutions of the corresponding linearized problem.

scholarly journals ON THE EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS FOR EQUATIONS DRIVEN BY THE $p$-LAPLACIAN AND WITH A NON-SMOOTH POTENTIAL

2003 ◽  
Vol 46 (1) ◽  
pp. 229-249 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Point Theory ◽  
Multiplicity Result ◽  
Primary 34C25 ◽  
Multiplicity Results ◽  
The Third ◽  
Smooth Potential ◽  
Periodic Problems ◽  
Semilinear Problems ◽  
Near Resonance ◽  
Multiple Periodic Solutions

AbstractIn this paper we examine periodic problems driven by the scalar $p$-Laplacian. Using non-smooth critical-point theory and a recent multiplicity result based on local linking (the original smooth version is due to Brezis and Nirenberg), we prove three multiplicity results, the third for semilinear problems with resonance at zero. We also study a quasilinear periodic eigenvalue problem with the parameter near resonance. We prove the existence of three distinct solutions, extending in this way a semilinear and smooth result of Mawhin and Schmitt.AMS 2000 Mathematics subject classification: Primary 34C25

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