# American Institute of Mathematical Sciences

## Journals

PROC
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:
CPAA
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:
CPAA
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:
CPAA
Communications on Pure & Applied Analysis 2019, 18(3): 1139-1154 doi: 10.3934/cpaa.2019055
We study positive solutions to (singular) boundary value problems of the form:
 \left\{ \begin{align} & -\left( {{\varphi }_{p}}(u') \right)'=\lambda h(t)\frac{f(u)}{{{u}^{\alpha }}},~\ \ t\in (0,1),~~ \\ & u'(1)+c(u(1))u(1)=0,~ \\ & u(0)=0, \\ \end{align} \right.
where
 $\varphi_p(u): = |u|^{p-2}u$
with
 $p>1$
is the
 $p$
-Laplacian operator of
 $u$
,
 $λ>0$
,
 $0≤α<1$
,
 $c:[0,∞)\rightarrow (0,∞)$
is continuous and
 $h:(0,1)\rightarrow (0,∞)$
is continuous and integrable. We assume that
 $f∈ C[0,∞)$
is such that
 $f(0)<0$
,
 $\lim_{s\rightarrow ∞}f(s) = ∞$
and
 $\frac{f(s)}{s^{α}}$
has a
 $p$
-sublinear growth at infinity, namely,
 $\lim_{s \rightarrow ∞}\frac{f(s)}{s^{p-1+α}} = 0$
. We will discuss nonexistence results for
 $λ≈ 0$
, and existence and uniqueness results for
 $λ \gg 1$
. We establish the existence result by a method of sub-supersolutions and the uniqueness result by establishing growth estimates for solutions.
keywords:
DCDS
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:
DCDS
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:
DCDS
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:
DCDS-S
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: