scholarly journals Continuum of one‐sign solutions of one‐dimensional Minkowski‐curvature problem with nonlinear boundary conditions

2021 ◽  
Author(s):  
Yanqiong Lu ◽  
Zhengqi Jing
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
One Dimensional ◽  
Curvature Problem

scholarly journals Continuum of one-sign solutions of one-dimensional Minkowski-curvature problem with nonlinear boundary conditions

2021 ◽  
Author(s):  
Yanqiong Lu ◽  
Zhengqi Jing
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Global Bifurcation ◽  
Nonlinear Boundary Conditions ◽  
Continuous Functions ◽  
One Dimensional ◽  
Curvature Equation ◽  
Existence And Multiplicity ◽  
Curvature Problem ◽  
The Continuum

In this work, we investigate the continuum of one-sign solutions of the nonlinear one-dimensional Minkowski-curvature equation $$-\big(u’/\sqrt{1-\kappa u’^2}\big)’=\lambda f(t,u),\ \ t\in(0,1)$$ with nonlinear boundary conditions $u(0)=\lambda g_1(u(0)), u(1)=\lambda g_2(u(1))$ by using unilateral global bifurcation techniques, where $\kappa>0$ is a constant, $\lambda>0$ is a parameter $g_1,g_2:[0,\infty)\to (0,\infty)$ are continuous functions and $f:[0,1]\times[-\frac{1}{\sqrt{\kappa}},\frac{1}{\sqrt{\kappa}}]\to\mathbb{R}$ is a continuous function. We prove the existence and multiplicity of one-sign solutions according to different asymptotic behaviors of nonlinearity near zero.

scholarly journals Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions

2019 ◽  
Vol 39 (5) ◽  
pp. 675-689
Author(s):  
D. D. Hai ◽  
X. Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Positive Parameter ◽  
One Dimensional ◽  
Existence Of Positive Solutions ◽  
The One

We prove the existence of positive solutions for the \(p\)-Laplacian problem \[\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}\] where \(\phi (s)=|s|^{p-2}s\), \(p\gt 1\), \(H_{i}:\mathbb{R}\rightarrow\mathbb{R}\) can be nonlinear, \(i=1,2\), \(f:(0,\infty )\rightarrow \mathbb{R}\) is \(p\)-superlinear or \(p\)-sublinear at \(\infty\) and is allowed be singular \((\pm\infty)\) at \(0\), and \(\lambda\) is a positive parameter.

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