Unilateral global bifurcation and nodal solutions for the pp-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 $(\mu_k^\nu(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $(\mathcal{C}_{k}^\nu)^+$ and $(\mathcal{C}_{k}^\nu)^-$, consisting of the bifurcation branch $\mathcal{C}_{k}^\nu$ from $(\mu_k^\nu(p), 0)$, where $\mu_k^\nu(p)$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $\nu\in\{+,-\}$. 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 $\lambda$ is a positive parameter, $f\in C([0,\infty),\mathbb{R})$ and for some $\beta>0$ such that $f(0)=0$, $f(s)>0$ for $s\in(\beta,\infty)$, $f(s)\beta)$ is the unique positive zero of $F(s)=\int_0^sf(t)\,dt$. In particular, we prove that there exist at least $2n+1$ positive solutions for $\lambda\in \big(\frac{n^2\pi^2}{f'(\beta)},\infty\big)$, where $n\in \mathbb{N}$. 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