ON SOME p(x)-KIRCHHOFF TYPE EQUATIONS WITH WEIGHTS
ON SOME p(x)-KIRCHHOFF TYPE EQUATIONS WITH WEIGHTS
Consider a class of p(x)-Kirchhoff type equations of the form <TEX>$$\left\{-M\left({\int}_{\Omega}\;\frac{1}{p(x)}{\mid}{\nabla}u{\mid}^{p(x)}\;dx\right)\;div\;({\mid}{\nabla}u{\mid}^{p(x)-2}{\nabla}u)={\lambda}V(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u=0\;on\;{\partial}{\Omega},$$</TEX> where p(x), <TEX>$q(x){\in}C({\bar{\Omega}})$</TEX> with 1 < <TEX>$p^-\;:=inf_{\Omega}\;p(x){\leq}p^+\;:=sup_{\Omega}p(x)$</TEX> < N, <TEX>$M:{\mathbb{R}}^+{\rightarrow}{\mathbb{R}}^+$</TEX> is a continuous function that may be degenerate at zero, <TEX>${\lambda}$</TEX> is a positive parameter. Using variational method, we obtain some existence and multiplicity results for such problem in …