We establish critical groups estimates for the weak solutions of $-{\mathrm{\Delta }}_{p}u=f(x,u)$ in Ω and $u=0$ on ∂Ω via Morse index, where Ω is a bounded domain, $f\in C^1(\bar{\mathrm{\Omega }}\times \mathbb{R})$ and $f(x,s)>0$ for all $x\in \bar{\mathrm{\Omega }}$, $s>0$ and $f(x,s)=0$ for all $x\in \bar{\mathrm{\Omega }}$, $s\le 0$. The proof relies on new uniform Sobolev inequalities for approximating problems. We also prove critical groups estimates when Ω is the ball or the annulus and f is a sign changing function.

中文翻译：

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拟线性椭圆方程的加权 Sobolev 空间和 Morse 估计

我们为弱解建立关键组估计$-{\mathrm{\Delta }}_p你=F（X,你）$单位为 Ω 和$你=0$通过莫尔斯指数计算 ∂Ω，其中 Ω 是有界域，$F\epsilon C^1（\bar{\mathrm{\Omega }}\times \mathbb{右}）$和$F（X,s）>0$对全部$X\epsilon \bar{\mathrm{\Omega }}$,$s>0$和$F（X,s）=0$对全部$X\epsilon \bar{\mathrm{\Omega }}$,$s\le 0$。该证明依赖于新的统一索博列夫不等式来近似问题。我们还证明了当 Ω 是球或环且f是变号函数时的临界群估计。