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 Bifurcation curves and exact multiplicity of positive solutions for Dirichlet problems with the Minkowski-curvature equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hongliang Gao ◽  
Jing Xu
Keyword(s):  
Positive Solutions ◽  
Dirichlet Problems ◽  
Precise Description ◽  
One Dimensional ◽  
Curvature Equation ◽  
Time Map ◽  
The One ◽  
Multiplicity Of Positive Solutions ◽  
Exact Multiplicity ◽  
Prime 2

AbstractIn this paper, we consider the bifurcation curves and exact multiplicity of positive solutions of the one-dimensional Minkowski-curvature equation $$ \textstyle\begin{cases} - (\frac{u'}{\sqrt{1-u^{\prime \,2}}} )'=\lambda f(u), &x\in (-L,L), \\ u(-L)=0=u(L), \end{cases} $$ { − ( u ′ 1 − u ′ 2 ) ′ = λ f ( u ) , x ∈ ( − L , L ) , u ( − L ) = 0 = u ( L ) , where λ and L are positive parameters, $f\in C[0,\infty ) \cap C^{2}(0,\infty )$ f ∈ C [ 0 , ∞ ) ∩ C 2 ( 0 , ∞ ) , and $f(u)>0$ f ( u ) > 0 for $0< u< L$ 0 < u < L . We give the precise description of the structure of the bifurcation curves and obtain the exact number of positive solutions of the above problem when f satisfies $f''(u)>0$ f ″ ( u ) > 0 and $uf'(u)\geq f(u)+\frac{1}{2}u^{2}f''(u)$ u f ′ ( u ) ≥ f ( u ) + 1 2 u 2 f ″ ( u ) for $0< u< L$ 0 < u < L . In two different cases, we obtain that the above problem has zero, exactly one, or exactly two positive solutions according to different ranges of λ. The arguments are based upon a detailed analysis of the time map.

scholarly journals Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

2007 ◽  
Vol 2007 ◽  
pp. 1-21
Author(s):  
Tsung-Fang Wu
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Ground State ◽  
Unbounded Domains ◽  
Ground State Solution ◽  
Multiple Positive Solutions ◽  
Dirichlet Problems ◽  
Existence And Multiplicity ◽  
Sobolev Constant ◽  
Multiplicity Of Positive Solutions

We consider the elliptic problem−Δu+u=b(x)|u|p−2u+h(x)inΩ,u∈H01(Ω), where2<p<(2N/(N−2)) (N≥3), 2<p<∞ (N=2), Ωis a smooth unbounded domain inℝN, b(x)∈C(Ω), andh(x)∈H−1(Ω). We use the shape of domainΩto prove that the above elliptic problem has a ground-state solution if the coefficientb(x)satisfiesb(x)→b∞>0as|x|→∞andb(x)≥cfor some suitable constantsc∈(0,b∞), andh(x)≡0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficientb(x)also satisfies the above conditions,h(x)≥0and0<‖h‖H−1<(p−2)(1/(p−1))(p−1)/(p−2)[bsupSp(Ω)]1/(2−p), whereS(Ω)is the best Sobolev constant of subcritical operator inH01(Ω)andbsup=supx∈Ωb(x).

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