PROC
Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents
Inbo Sim Yun-Ho Kim
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
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

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