论文标题
涉及强大潜力的双期问题
A double phase problem involving Hardy potentials
论文作者
论文摘要
在本文中，我们处理以下双相问题$$ \ left \ weet {arnay} {ll} {ll} - \ mbox {div} \ left（| \ nabla u |^{p-2} \ nabla u+a（x） γ\左（\ displayStyle \ frac {| u |^{p-2} u} {| x |^|^p}+a（x）\ displayStyle \ frac {| u | u | u |^{q-2} u} } \partialΩ，\ end {array} \ right。 $$，其中$ω\ subset \ mathbb r^n $是一个开放式的，带有lipschitz边界的开放式，$ 0 \inΩ$，$ n \ geq2 $，$ 1 <p <q <n $，weight $ a（\ cdot）\ geq0 $，$γ$，$γ$是一个真实的参数，$ f $是一个subcritation carlitation carlitation foruct。通过变异方法，我们在Musielak-Orlicz-Sobolev空间$ W^{1，\ Mathcal H} _0（ω）$上提供了非平凡的弱解，带有模块化函数$ \ MATHCAL H（t，t，x）= t^p+a（x）t^q $。为此，我们首先在$ a（\ cdot）$上的合适假设下，首先引入空间$ w^{1，\ mathcal H} _0（ω）$的Hardy不平等现象。
In this paper, we deal with the following double phase problem $$ \left\{\begin{array}{ll} -\mbox{div}\left(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right)= γ\left(\displaystyle\frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle\frac{|u|^{q-2}u}{|x|^q}\right)+f(x,u) & \mbox{in } Ω,\\ u=0 & \mbox{in } \partialΩ, \end{array} \right. $$ where $Ω\subset\mathbb R^N$ is an open, bounded set with Lipschitz boundary, $0\inΩ$, $N\geq2$, $1<p<q<N$, weight $a(\cdot)\geq0$, $γ$ is a real parameter and $f$ is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space $W^{1,\mathcal H}_0(Ω)$, with modular function $\mathcal H(t,x)=t^p+a(x)t^q$. For this, we first introduce the Hardy inequalities for space $W^{1,\mathcal H}_0(Ω)$, under suitable assumptions on $a(\cdot)$.
