PROC
Semilinear elliptic equations with generalized cubic nonlinearities
Junping Shi R. Shivaji
Conference Publications 2005, 2005(Special): 798-805 doi: 10.3934/proc.2005.2005.798
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
Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions
Dagny Butler Eunkyung Ko Eun Kyoung Lee R. Shivaji
Communications on Pure & Applied Analysis 2014, 13(6): 2713-2731 doi: 10.3934/cpaa.2014.13.2713
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
CPAA
Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system
Mohan Mallick R. Shivaji Byungjae Son S. Sundar
Communications on Pure & Applied Analysis 2018, 17(3): 1295-1304 doi: 10.3934/cpaa.2018062
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
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 5153-5166 doi: 10.3934/dcds.2013.33.5153
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
Discrete & Continuous Dynamical Systems - A 2003, 9(4): 1063-1071 doi: 10.3934/dcds.2003.9.1063
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
Discrete & Continuous Dynamical Systems - A 2010, 27(3): 1123-1132 doi: 10.3934/dcds.2010.27.1123
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
Discrete & Continuous Dynamical Systems - S 2014, 7(6): 1181-1191 doi: 10.3934/dcdss.2014.7.1181
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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