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Search: All articles in the CJM digital archive with keyword positive solution

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1. CJM Online first

Du, Zhuoran; Fang, Yanqin; Gui, Changfeng
A class of degenerate elliptic equations with nonlinear boundary conditions
We consider positive solutions of the problem \begin{equation} (*)\qquad \left\{ \begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=bx_{n}^{a}u^{p}\;\;\;\;\;\mbox{in}\;\;\mathbb{R}_{+}^{n}, \\ \frac{\partial u}{\partial \nu^a}=u^{q} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{on}\;\;\partial \mathbb{R}_{+}^{n}, \\ \end{array} \right. \end{equation} where $a\in (-1,0)\cup(0,1)$, $b\geq 0$, $p, q\gt 1$ and $\frac{\partial u}{\partial \nu^a}:=-\lim_{x_{n}\rightarrow 0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. In special case $b=0$, it is associated to fractional Laplacian equation $(-\Delta)^{s}u=u^{q} $ in entire space $\mathbb{R}^{n-1}$. We obtain the existence of positive axially symmetric solutions to ($*$) for the case $a\in (-1,0)$ in $n\geq3$ for supercritical exponents $p\geq\frac{n+a+2}{n+a-2}, \;\;q\geq\frac{n-a}{n+a-2}$. The nonexistence is obtained for the case $a\in (-1,0)$, $b\geq 0$ and any $p,~q\gt 1$ in $n=2$ as well.

Keywords:existence, non-existence, positive solutions, degenerate elliptic equation, nonlinear boundary conditions, symmetry, monotonicity
Categories:35D30, 35J70, 35J25

2. CJM 2006 (vol 58 pp. 449)

Agarwal, Ravi P.; Cao, Daomin; Lü, Haishen; O'Regan, Donal
Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$-Laplacian Equations
Positive solutions are obtained for the boundary value problem \[\begin{cases} -( | u'| ^{p-2}u')' =\lambda f( t,u),\;t\in ( 0,1) ,p>1\\ u( 0) =u(1) =0. \end{cases} \] Here $f(t,u) \geq -M,$ ($M$ is a positive constant) for $(t,u) \in [0\mathinner{,}1] \times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper-lower solution method.

Keywords:one dimensional $p$-Laplacian, positive solution, degree theory, upper and lower solution

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