151 research outputs found
Unilateral global bifurcation and nodal solutions for the -Laplacian with sign-changing weight
In this paper, we shall establish a Dancer-type unilateral global bifurcation
result for a class of quasilinear elliptic problems with sign-changing weight.
Under some natural hypotheses on perturbation function, we show that
is a bifurcation point of the above problems and there are
two distinct unbounded continua, and
, consisting of the bifurcation branch
from , where is the
-th positive or negative eigenvalue of the linear problem corresponding to
the above problems, . As the applications of the above
unilateral global bifurcation result, we study the existence of nodal solutions
for a class of quasilinear elliptic problems with sign-changing weight.
Moreover, based on the bifurcation result of Dr\'{a}bek and Huang (1997)
[\ref{DH}], we study the existence of one-sign solutions for a class of high
dimensional quasilinear elliptic problems with sign-changing weight
Positive solutions of a nonlinear m-point boundary value problem
AbstractLet ai ≥ 0 for i = 1,…, m − 3 and am−2 > 0. Let ξi satisfy 0 < ξ1 < ξ2 < … < ξm−2 < 1 and Σm−2i=1 aiξi < 1. We study the existence of positive solutions to the boundary-value problem where a ϵ C([0, 1], [0, ∞)), and f ϵ C([0, ∞), [0, ∞)). We show the existence of at least one positive solution if f is either superlinear or sublinear by applying a fixed-point theorem in cones
Multiplicity of positive solutions for second-order three-point boundary value problems
AbstractWe study the multiplicity of positive solutions for the second-order three-point boundary value problem u″+λh(t)f(u)=0, t∈(0,1), u(0)=0, αu(η)=u(1)where η: 0 < η < 1, 0 < α < 1η. The methods employed are fixed-point index theorems and Leray-Schauder degree and upper and lower solutions
Multiple positive solutions for a class of Neumann problems
We study the existence of multiple positive solutions of the Neumann problem
\begin{equation*}
\begin{split}
-u''(x)&=\lambda f(u(x)), \qquad x\in(0,1),\\
u'(0)&=0=u'(1),
\end{split}
\end{equation*}
where is a positive parameter, and for some such that , for , is the unique positive zero of . In particular, we prove that there exist at least positive solutions for , where . The proof of our main result is based upon the bifurcation and continuation methods
Positive solutions for nonlinear m-point boundary value problems of dirichlet type via fixed-point index theory
AbstractLet a ϵ C[0,1], b ϵ C([0,1], (-∞, 0)). Let φ1(t) be the unique solution of the linear boundary value problem u″(t)+s(t)u′(t)+b(t)u(t)=0, tϵ(0,1),u(0)=0, u(1)=1. We study the multiplicity of positive solutions for the m-point boundary value problems of Dirichlet type u″+a(t)u′+b(t)u+g(t)f(u)=0,u(0)=0, u(1)−∑i=1m−2αiu(ξi)=0, where ξi ϵ (0,1) and αi ϵ (0, ∞), i ϵ {… , m−2), are given constants satisfying Σi=1m−1 αiφ1(ξi) < 1. The methods employed are fixed-point index theory
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