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We study positive solutions to the two-point boundary value problem: $$\begin{aligned} \begin{matrix} -u''=\lambda h(t) f(u)~;~(0,1) \\ u(0)=0\\ u'(1)+c(u(1))u(1)=0,\end{matrix} \end{aligned}$$ where $$\lambda >0$$ is a parameter, $$h \in C^1((0,1],(0,\infty ))$$ is a decreasing function, $$f \in C^1((0,\infty ),\mathbb {R}) $$ is an increasing concave function such that $$\lim \limits _{s \rightarrow \infty }f(s)=\infty $$ , $$\lim \limits _{s \rightarrow \infty }\frac{f(s)}{s}=0$$ , $$\lim \limits _{s \rightarrow 0^+}f(s)=-\infty $$ (infinite semipositone) and $$c \in C([0,\infty ),(0,\infty ))$$ is an increasing function. For classes of such h and f, we establish the uniqueness of positive solutions for $$\lambda \gg 1$$ .

We prove the uniqueness of positive radial solutions to a class of singular p-Laplacian equations in a ball with Dirichlet boundary condition when a parameter is large. The reaction term exhibits i...

We prove the existence of positive classical solutions for the \begin{document}$ p $\end{document} -Laplacian problem \begin{document}$ \begin{equation*} \left\{ \begin{array}{c} -(r(t)\phi (u^{\prime }))^{\prime } = -\frac{\lambda }{u^{\delta }}+f(t,u),\ t\in (0,1), \\ u(0) = u(1) = 0,\end{array}\right. \end{equation*} $\end{document} where \begin{document}$ 0 , \begin{document}$ \phi (s) = |s|^{p-2}s $\end{document} , \begin{document}$ p>1 $\end{document} , \begin{document}$ f:(0,1)\times \lbrack 0,\infty )\rightarrow \mathbb{R} $\end{document} is a Caratheodory function satisfying \begin{document}$ \limsup\limits_{z\rightarrow 0^{+}}\frac{f(t,z)}{z^{p-1}} uniformly for a.e. \begin{document}$ t $\end{document} \begin{document}$ \in (0,1), $\end{document} where \begin{document}$ \lambda_{1} $\end{document} denotes the principal eigenvalue of \begin{document}$ -(r(t)\phi (u^{\prime }))^{\prime } $\end{document} with zero boundary conditions, and \begin{document}$ \lambda $\end{document} is a small nonnegative parameter.

We prove the existence of positive radial solutions to the problem \begin{document}$ \begin{cases} -\Delta _{p}u = \lambda \ K(|x|)f(u)\ \text{in } |x|>r_{0}, \\ \dfrac{\partial u}{\partial n}+\tilde{c}(u)u = 0\ \text{on }|x| = r_{0},\ \ u(x)\rightarrow 0\text{ as }|x|\rightarrow \infty ,\end{cases} $\end{document} where \begin{document}$ \ \Delta _{p}u = div(|\nabla u|^{p-2}\nabla u),\ N>p>1, \Omega = \{x\in \mathbb{R}^{N}:|x|>r_{0}>0\}, $\end{document} \begin{document}$ f:(0,\infty )\rightarrow \mathbb{R} $\end{document} is \begin{document}$ p $\end{document} -superlinear at \begin{document}$ \infty $\end{document} with possible singularity at \begin{document}$ 0, $\end{document} and \begin{document}$ \lambda $\end{document} is a small positive parameter. A nonexistence result is also established when \begin{document}$ f $\end{document} has semipositone structure at \begin{document}$ 0. $\end{document}

We prove the existence and nonexistence of positive solutions to the p-Laplacian problem −Δpu=λm(x)f(u) in Ω,u=0 on ∂Ω, where Ω is a bounded domain in Rn with smooth boundary ∂Ω, f(u)∼uq at ∞ for s...

We prove the existence of positive solutions for the boundary value problem $$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$ for certain range of the parameter $$\lambda >0$$ , where $$m\in (1/2,1/2+\varepsilon )$$ with $$\varepsilon >0$$ small, and f is superlinear or sublinear at $$\infty $$ with no sign-conditions at 0 assumed.

We prove the $$C^{1,\beta }$$ -boundary regularity and a comparison principle for weak solutions of the problem $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u-\lambda \psi _{p}(u)=f(x)&{}\quad \text {in }\Omega , \\ u=0&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a bounded domain in $$\mathbb {R}^{N},N>1\ $$ with smooth boundary $$\partial \Omega ,\ \ \Delta _{p}u=\mathrm{div}(|\nabla u|^{p-2}\nabla u),\psi _{p}(u)=|u|^{p-2}u,p>1,\ $$ and f is allowed to be unbounded.

We prove the existence of a positive classical solution for the p-Laplacian equation–(r(t)ϕ(u'))'= –λh(u) + f (t, u), t ∈ (0, 1)with Sturm-Liouville boundary conditions, whereϕ(s)= |s|p‒2s;p> 1;r: [0, 1] → (0, ∞); f : (0, 1) × [0;∞) →ℝis a Carathéodory function satisfying a superlinear condition at 0 and 1 involving the principal eigenvalue of –(r(t)ϕ(u'))'h: (0,∞) → (0,∞) is allowed to have infinite semipositone structure at 0, andλ≥ 0 is a small parameter.

We prove uniqueness of positive solutions for the boundary value problem \begin{equation*} \left\{ \begin{array}{l} -\Delta u=\lambda f(u)\text{ in }\Omega , \\ \ \ \ \ \ \ \ u=0\text{ on }\partial \Omega , \end{array} \right. \end{equation*} where Ω is a bounded domain in $\mathbb{R}$n with smooth boundary ∂ Ω, λ is a large positive parameter, f:(0,∞) → [0,∞) is nonincreasing for large t and is allowed to be singular at 0.

We prove the existence of positive ω-periodic solutions for the delayed differential equation $$x^{\prime}(t) = a(t)g(x(t))x(t) - \lambda b(t)f(x(t - \tau (t))),$$ where λ is a positive parameter, \({a,b,\tau \in C(\mathbb{R},\mathbb{R})}\) are ω-periodic functions with \({a,b\geq 0,a,b \not \equiv 0,f,g\in C([0,\infty ),[0,\infty ))}\), g does not need to be bounded above or bounded away from 0, and g(0) = 0 is allowed.

We prove the existence and nonexistence of positive radial solutions for the system $$\left\{\begin{array}{ll} -\Delta_{p}u_{1}=h_{1}(u_{2})+\mu _{1}f_{1}(u_{2}) & \quad \text{in} \, B, \\ -\Delta_{p}u_{2}=h_{2}(u_{1})+\mu _{2}f_{2}(u_{1}) & \quad \text{in}\, B, \\ u_{1}=u_{2}=0 & \quad \text{on} \, \partial B, \end{array}\right.$$ where \({p > 1, \Delta _{p}u = {\rm div}(|\nabla u|^{p-2}\nabla u), \, B}\) is the open unit ball in\({\mathbb{R}^{N},h_{i}, f_{i}:(0,\infty) \rightarrow \mathbb{R}}\) with fi asymptotically p-linear at ∞, and μi are positive constants, i = 1, 2.

We prove the nonexistence of positive solutions to the system $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u=\lambda f(v)\text { in }B, \\ -\Delta v=\mu g(u) \text { in }B, \\ u=v=0\text { on }\partial B, \end{array} \right. \end{aligned}$$ where $$B$$ is the open unit ball in $$\mathbb {R}^{N}$$ , $$\ N>1,\ \lambda ,\mu $$ are positive constants bounded away from $$0$$ with $$\lambda \mu $$ large, $$f,g$$ are smooth functions with $$f(0),g(0)

We prove uniqueness of positive solutions for the boundary value problems \[ \{\begin{array}{ll} -\Delta u=\lambda f(u)\ \ &\text{in}\Omega, \ \ \ \ \ u=0 &\text{on \partial \Omega, \] where Ω is a bounded domain in ℝn with smooth boundary ∂Ω, λ is a positive parameter and f:(0,∞) → (0,∞) is sublinear at ∞ and is allowed to be singular at 0.

We consider the existence of positive solutions for the boundary-value problem(q(t)ϕ(u′))′ + λf(t,u) = 0, r < t < R,au(r) − bϕ−1 (q(r))u′(r) = 0, cu(R) + dϕ−1(q(R))u′(R) = 0,where ϕ(u′) = |u′|p−2u′, p > 1, λ > 0, f is p-superlinear or p-sublinear at ∞ and is allowed to become −∞ at u = 0. Our results unify and extend many known results in the literature.

(2008). An Elementary Proof of the Error Estimates in Simpson's Rule. Mathematics Magazine: Vol. 81, No. 4, pp. 295-300.

We establish existence results for positive solutions of the boundary value problem$¥left¥{ ¥begin{array}{l} (q(t)¥phi (u^{¥prime }))^{¥prime }+¥lambda r(t)f(u)=0,¥;t¥in (a,b) ¥¥ u(a)=u(b)=0,¥end{array}¥right.$where φ(x) = |x|p - 2 x, p > 1, q ∈ C[a,b], r ∈ L1 (a,b), f is allowed to have negative values and become infinite at 0, and λ is a positive parameter.

We prove the uniqueness of positive solutions for a class of second-order nonlinear elliptic boundary-value problems with sublinear nonlinearities.

We obtain necessary and sufficient conditions for the existence of positive solutions for a class of sublinear Dirichlet quasilinear elliptic systems.

Consider the system where λ is a positive parameter and Ω is a bounded domain in RN. We prove the existence of a large positive solution for λ large when limx → ∞ (f(Mg(x))/x) = 0 for every M > 0. In particular, we do not need any monotonicity assumptions on f, g, nor any sign conditions on f(0), g(0).

We establish existence and multiplicity of positive solutions to the quasilinear boundary value problem div(|⊇u| p-2 ⊇ u ) = -λf(u) in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in R n with smooth boundary ∂Ω, f: [0, ∞) → R is continuous and p-sublinear at ∞, and A is a large parameter.

SynopsisWe consider the existence of positive solutions to a class of singular nonlinear boundary value problems for P-Laplacian-like equations. Our approach is based on the Schauder Fixed-Point Theorem.

This paper is concerned with existence and uniqueness results for fourth order boundary value problems in a Hilbert space. Our approach is based on the Leray-Schauder continuation theorem.

We prove the existence of\ positive solutions for the $p$-Laplacian system \[ \left\{ \begin{array}{@{\,}c@{\enskip}c} -\Delta _{p}u_{1}=\lambda f_{1}(u_{2}) &\text{in}~\Omega \,, \\ -\Delta _{p}u_{2}=\lambda f_{2}(u_{1}) &\text{in}~\Omega \,, \\ \ \ \ \quad u_{1}=u_{2}=0 & \text{on}~\partial \Omega \,, \end{array} \right. \] where $\Delta _{p}u=\mbox{div}(|\nabla u|^{p-2}\nabla u),p>1, \Omega$ is a bounded domain in $\mathbf{R}^{n}$ with smooth boundary $\partial \Omega ,f_{i}:(0,\infty) \rightarrow \mathbf{R}$ are possibly singular at 0 and are not required to be positive or nondecreasing, and $\lambda $ is a large parameter.

In this paper we study positive solutions for the system \begin{align*} -(r^{N-1}\phi(u'))' & \, =\, \lambda r^{N-1}f(v);\ a\, 0 $ is a parameter and $ \phi $ is an odd, increasing homeomorphism on $ \Bbb R $. Here $ f,\, g \in C[0,\infty) $ belong to a class of superlinear functions at $\infty$. In particular we allow $ f(0) $ or $ g(0) $ or both to be negative (semipositone system). We discuss the existence of a positive solution for $ \lambda $ small. Our proof is based on degree theory.

In this paper we establish the existence of positive radial solutions for a class of quasilinear elliptic boundary problems. Both the positone and nonpositone cases are considered.

We study positive solutions for the system $-\Delta_p u = \lambda f(v)$ in $\quad \Omega $ $-\Delta_p v = \lambda g(u)$ in $ \quad \Omega $ $u = 0 = v$ on $ \quad \partial \Omega$ where $ \lambda > 0 $ is a parameter, $ \Delta_p $ denotes the p-Laplacian operator defined by $ \Delta_p(z)$:=div$(|\nabla z|^{p-2}\nabla z) $ for $ p> 1 $ and $ \Omega $ is a bounded domain with smooth boundary. Here $ f,g \in C[0,\infty) $ belong to a class of functions satisfying $ \lim_{z \to \infty}\frac{f(z)}{z^{p-1}}=0, \lim_{z \to \infty}\frac{g(z)}{z^{p-1}}=0 $. In particular, we discuss the existence of radial solutions for large $ \lambda $ when $ \Omega $ is an annulus. For a general bounded region $ \Omega, $ we also discuss a non-existence result when $ f(0) < 0 $ and $ g(0) < 0. $