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http://dx.doi.org/10.14317/jami.2014.113

ON SOME p(x)-KIRCHHOFF TYPE EQUATIONS WITH WEIGHTS  

Chung, Nguyen Thanh (Dep. Science Management & International Cooperation, Quang Binh University)
Publication Information
Journal of applied mathematics & informatics / v.32, no.1_2, 2014 , pp. 113-128 More about this Journal
Abstract
Consider a class of p(x)-Kirchhoff type equations of the form $$\left\{-M\left({\int}_{\Omega}\;\frac{1}{p(x)}{\mid}{\nabla}u{\mid}^{p(x)}\;dx\right)\;div\;({\mid}{\nabla}u{\mid}^{p(x)-2}{\nabla}u)={\lambda}V(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u=0\;on\;{\partial}{\Omega},$$ where p(x), $q(x){\in}C({\bar{\Omega}})$ with 1 < $p^-\;:=inf_{\Omega}\;p(x){\leq}p^+\;:=sup_{\Omega}p(x)$ < N, $M:{\mathbb{R}}^+{\rightarrow}{\mathbb{R}}^+$ is a continuous function that may be degenerate at zero, ${\lambda}$ is a positive parameter. Using variational method, we obtain some existence and multiplicity results for such problem in two cases when the weight function V (x) may change sign or not.
Keywords
p(x)-Kirchhoff type equations; Mountain pass theorem; Ekeland variational principle; Weight functions;
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