PROC
Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents
Inbo Sim Yun-Ho Kim
Conference Publications 2013, 2013(special): 695-707 doi: 10.3934/proc.2013.2013.695
We study the following nonlinear problem \begin{equation*} -div(w(x)|\nabla u|^{p(x)-2}\nabla u)=\lambda f(x,u)\quad in \Omega \end{equation*} which is subject to Dirichlet boundary condition. Under suitable conditions on $w$ and $f$, employing the variational methods, we show the existence of solutions for the above problem in the weighted variable exponent Lebesgue-Sobolev spaces. Also we obtain the positivity of the infimum eigenvalue for the problem.
keywords: weighted variable exponent Lebesgue-Sobolev spaces fountain theorem mountain pass theorem $p(x)$-Laplacian eigenvalue.
CPAA
On the existence of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition
Inbo Sim
Communications on Pure & Applied Analysis 2008, 7(4): 905-923 doi: 10.3934/cpaa.2008.7.905
We obtain the existence results of nodal solutions for singular one-dimensional $\varphi$-Laplacian problem with asymptotic condition:

$\varphi (u'(t))' + \lambda h(t) f (u(t)) = 0,\ \ $ a.e. $\ t \in (0,1), \qquad\qquad\qquad\qquad\qquad $ $(\Phi_\lambda)$

$u(0) = 0=u(1),$

where $\varphi : \mathbb R \to \mathbb R$ is an odd increasing homeomorphism, $\lambda$ a positive parameter and $h \in L^1(0,1)$ a nonnegative measurable function on $(0,1)$ which may be singular at $t = 0$ and/or $t = 1,$ and $f \in C(\mathbb R, \mathbb R)$ and is odd.

keywords: existence global bifurcation asymptotic condition. Singular one-dimensional $\varphi$-Laplacian
CPAA
Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions
Eun Kyoung Lee R. Shivaji Inbo Sim Byungjae Son
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
$u$
,
$λ>0$
,
$0≤α<1$
,
$c:[0,∞)\rightarrow or \to(0,∞)$
is continuous and
$h:(0,1)\rightarrow or \to(0,∞)$
is continuous and integrable. We assume that
$f∈ C[0,∞)$
is such that
$f(0)<0$
,
$\lim_{s\rightarrow or \to∞}f(s) = ∞$
and
$\frac{f(s)}{s^{α}}$
has a
$p$
-sublinear growth at infinity, namely,
$\lim_{s \rightarrow or \to ∞}\frac{f(s)}{s^{p-1+α}} = 0$
. 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: Semipositone p-Laplacian p-sublinear growth at infinity nonlinear boundary conditions positive solutions existence uniqueness

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