CPAA
Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions
Dagny Butler Eunkyung Ko Eun Kyoung Lee R. Shivaji
We study positive radial solutions to the boundary value problem \begin{eqnarray} -\Delta u = \lambda K(|x|)f(u), \quad x \in \Omega, \\ \frac{\partial u}{\partial \eta}+\tilde{c}(u)u = 0, \quad |x|=r_0, \\ u(x) \rightarrow 0, \quad |x|\rightarrow \infty, \end{eqnarray} where $\Delta u=div \big(\nabla u\big)$ is the Laplacian of $u$, $\lambda$ is a positive parameter, $\Omega=\{x \in \mathbb{R}^N| N>2, |x|> r_0 \mbox{ with }r_0>0\}$, $K:[r_0, \infty)\rightarrow(0,\infty)$ is a continuous function such that $\lim_{r \rightarrow \infty}K(r)=0$, $\frac{\partial}{\partial \eta}$ is the outward normal derivative, and $\tilde{c}:[0,\infty) \rightarrow (0,\infty)$ is a continuous function. We consider various $C^1$ classes of the reaction term $f:[0,\infty) \rightarrow \mathbb{R}$ that are sublinear at $\infty$ $(i.e. \lim_{s \rightarrow \infty}\frac{f(s)}{s}=0)$. In particular, we discuss existence and multiplicity results for classes of $f$ with $(a)$ $f(0)>0$, $(b)$ $f(0)<0$, and $(c)$ $f(0)=0$. We establish our existence and multiplicity results via the method of sub-super solutions. We also discuss some uniqueness results.
keywords: nonlinear boundary conditions exterior domains. Positive solution
PROC
Semilinear elliptic equations with generalized cubic nonlinearities
Junping Shi R. Shivaji
A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.
keywords: global bifurcation. semilinear reaction-diffusion equation
CPAA
Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system
Mohan Mallick R. Shivaji Byungjae Son S. Sundar
In this paper we study the positive solutions to the
$n\times n$
$p$
-Laplacian system:
$\begin{equation*}\begin{cases}-\left(\varphi_{p_1}(u_1')\right)' = \lambda h_1(t) \left(u_1^{p_1-1-\alpha_1}+f_1(u_2)\right),\quad t\in (0,1),\\-\left(\varphi_{p_2}(u_2')\right)' = \lambda h_2(t) \left(u_2^{p_2-1-\alpha_2}+f_2(u_3)\right),\quad t\in (0,1),\\\quad\quad\quad\vdots\qquad\,\: =\quad\quad\quad\quad\quad\quad \vdots\\-\left(\varphi_{p_n}(u_n')\right)' = \lambda h_n(t) \left(u_n^{p_n-1-\alpha_n}+f_n(u_1)\right),~~\, t\in (0,1),\\\quad\,\,\,\, u_j(0)=0=u_j(1); ~~ j=1,2,\dots,n, \\ \end{cases}\end{equation*}$
where
$\lambda$
is a positive parameter,
$p_j>1$
,
$\alpha_j\in(0,p_j-1)$
,
$\varphi_{p_j}(w)=|w|^{p_j-2}w$
, and
$h_j \in C((0,1),(0, \infty))\cap L^1((0,1),(0,\infty))$
for
$j=1,2,\dots,n$
. Here
$f_j:[0,\infty)\rightarrow[0,\infty)$
,
$j=1,2,\dots,n$
are nontrivial nondecreasing continuous functions with
$f_j(0)=0$
and satisfy a combined sublinear condition at infinity. We discuss here a bifurcation result, an existence result for
$\lambda>0$
, and a multiplicity result for a certain range of
$\lambda$
. We establish our results through the method of sub-super solutions.
keywords: \begin{document}$p$\end{document}-Laplacian singular system positive solution multiplicity and bifurcation
DCDS
Multiplicity results for classes of singular problems on an exterior domain
Eunkyoung Ko Eun Kyoung Lee R. Shivaji
We study radial positive solutions to the singular boundary value problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda K(|x|)\frac{f(u)}{u^\beta} \quad \mbox{in}~ \Omega,\\ ~~~u(x) = 0 \qquad \qquad \qquad \qquad~~\mbox{if}~|x|=r_0,\\ ~~~u(x) \rightarrow 0 \qquad\qquad \qquad \mbox{if}~|x|\rightarrow \infty, \end{cases} \end{equation*} where $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, $1 < p < N, N >2, \lambda > 0, 0 \leq \beta <1 ,\Omega= \{ x \in \mathbb{R}^{N} : |x| > r_0 \}$ and $ r_0 >0$. Here $f:[0, \infty)\rightarrow (0, \infty)$ is a continuous nondecreasing function such that $\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0$ and $ K \in C( (r_0, \infty),(0, \infty) ) $ is such that $\int_{r_0}^{\infty} r^\mu K(r) dr < \infty, $ for some $\mu > p-1$. We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha+u}}$ for $ \alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-super solutions.
keywords: exterior domain Singular p-Laplacain problem multilpicity sub-supersolution. positive radial solution
DCDS
On positive solutions for classes of p-Laplacian semipositone systems
Maya Chhetri D. D. Hai R. Shivaji
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. $

keywords: p-Laplacian systems positive solutions.
DCDS
Classes of singular $pq-$Laplacian semipositone systems
Eun Kyoung Lee R. Shivaji Jinglong Ye
We consider the positive solutions to classes of $pq-$Laplacian semipositone systems with Dirichlet boundary conditions, in particular, we study strongly coupled reaction terms which tend to $-\infty$ at the origin and satisfy a combined sublinear condition at $\infty.$ By using the method of sub-super solutions we establish our results.
keywords: singular semipositone system sub-super solutions positive solutions.
DCDS-S
Alternate steady states for classes of reaction diffusion models on exterior domains
Dagny Butler Eunkyung Ko R. Shivaji
We study positive radial solutions to the problem \begin{equation*} \left\{ \begin{split} -\Delta u &= \lambda K(|x|)f(u), \quad x \in \Omega, \\u(x) &= 0 \qquad
   
   
    \mbox{ if } |x|=r_0, \\u(x) &\rightarrow 0 \qquad
   
    \mbox{ as } |x|\rightarrow\infty, \end{split} \right. \end{equation*} where $\Delta u=div \big(\nabla u\big)$ is the Laplacian of $u$, $\lambda$ is a positive parameter, $\Omega=\{x\in\mathbb{R}^N: |x|>r_0\}$, $r_0>0$, and $N>2$. Here, $f\in C^2[0,\infty)$ and $f(u)>0$ on $(0,\sigma)$ and $f(u)<0$ for $u>\sigma$. Furthermore, $K:[r_0, \infty)\rightarrow(0,\infty)$ is continuous and $\lim_{r\rightarrow\infty}K(r)=0$. We discuss the existence of multiple positive solutions for a certain range of $\lambda$ leading to the occurrence of an S-shaped bifurcation curve when $f$ satisfies some additional assumptions. In particular, the two models we consider are $f_1(u)=u-\frac{u^2}{K}-c\frac{u^2}{1+u^2}$ and $f_2(u)=\tilde{K}-u+\tilde{c}\frac{u^4}{1+u^4}$. We prove our results by the method of sub-super solutions.
keywords: Steady states reaction diffusion models logistic models with grazing exterior domains phosphorus cycling models.

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