On an anisotropic problem with singular nonlinearity having variable exponent
Tóm tắt
We consider the following anisotropic problem, with singular nonlinearity having a variable exponent
$$\begin{aligned} \left\{ \begin{array}{ll} -\sum \limits _{i=1}^{N}\partial _{i}\left[ \left| \partial _{i}u\right| ^{p_{i}-2}\partial _{i}u\right] =\frac{f}{u^{\gamma (x) }} &{} \quad in~\Omega , \\ u=0 &{} \quad on~\Omega , \\ u\ge 0 &{} \quad in~\Omega ; \end{array} \right. \end{aligned}$$
where
$$\Omega $$
is a bounded regular domain in
$${\mathbb {R}}^{N}$$
and
$$\gamma (x)>0$$
is a smooth function, having a convenient behavior near
$$\partial \Omega .$$
f is assumed to be a non negative function belonging to a suitable Lebesgue space
$$L^{m}\left( \Omega \right) .$$
We will also assume without loss of generality that
$$2\le p_{1}\le p_{2}\le \cdots \le p_{N}.$$
Using approximation techniques, we obtain existence and regularity of positive solutions to the considered problem.
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