Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains
Chao Zhang Lihe Wang Shulin Zhou Yun-Ho Kim
Communications on Pure & Applied Analysis 2014, 13(6): 2559-2587 doi: 10.3934/cpaa.2014.13.2559
In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Caldéron--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.
keywords: Elliptic Gradient estimate $p(x)$-Laplacian Reifenberg domain. BMO space
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.
Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition
Eun Bee Choi Yun-Ho Kim
Conference Publications 2015, 2015(special): 276-286 doi: 10.3934/proc.2015.0276
We study the following elliptic equations with variable exponents \begin{equation*} \begin{cases} -\text{div}(\varphi(x,\nabla u))+{|u|}^{p(x)-2}u= f(x,u) \quad &\text{in } \Omega \\ \varphi(x,\nabla u) \frac{\partial u}{\partial n}= g(x,u) & \text{on }\partial\Omega. \end{cases} \tag{P} \end{equation*} Under suitable conditions on $\phi$, $f$, and $g$, by employing the mountain pass theorem, the problem (P) has at least one nontrivial weak solution without assuming the Ambrosetti and Rabinowitz type condition.
keywords: p(x)-Laplace type Mountain pass theorem Nonlinear boundary conditions. Variable exponent Sobolev spaces Weak solution

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