论文标题
具有凹形和奇异非线性的$ p-q $ laplacian系统的解决方案的存在和多样性
Existence and multiplicity of solutions to a $p-q$ Laplacian system with a concave and singular nonlinearities
论文作者
论文摘要
在本文中，我们研究了以下问题系统的多种非平凡积极弱解决方案的存在。 \ begin {align*} \ begin {split}-Δ_{p} u-δ_qu＆=λf（x）| u |^{r-2} u+++n+n+n+frac {1-α} {2-α-β} {2-α-β} h（x）h（x） | U |^{ - α} | V |^{1-β} \，\，\，\ Mbox {in} \，\，\，\，\\ - \ \ - \ \ - \ \ - \ \ - \ \ - {p} v-Δ_qv＆=μg（x）| v |^{r-2} | u |^{1-α} | V |^{ - β} \，\，\，\ mbox {in} \，\，\，ω，\\ u，v＆> 0 \，\，\，\，\ mbox {in} \partialΩ\ end {split} \ end {align*}其中（c）：〜$ 0 <α<1，\; 0 <β<1，$ $2-α-β<q <q <q <q <\ frac {n（p-1）} {n-p} {n-p} <p <p <p <p <p <p <p^*$，with $ p^*$，$ p^*$，$ p^*= \ frac = \ frac} np} $} np}我们将保证在Nehari歧管中存在解决方案。此外，通过使用Lusternik-Schnirelman类别，我们将证明至少存在$ \ text {cat}（ω）+1 $ $ $。
In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -Δ_{p}u-Δ_q u &= λf(x)|u|^{r-2}u+ν\frac{1-α}{2-α-β}h(x) |u|^{-α}|v|^{1-β}\,\,\mbox{in}\,\,Ω,\\ -Δ_{p}v-Δ_q v &= μg(x)|v|^{r-2}v+ν\frac{1-β}{2-α-β}h(x) |u|^{1-α}|v|^{-β}\,\,\mbox{in}\,\,Ω,\\ u,v&>0\,\,\mbox{in}\,\,Ω,\\ u= v &= 0\,\, \mbox{on}\,\, \partialΩ\end{split} \end{align*} where (C):~$0<α<1,\;0<β<1,$ $2-α-β<q<\frac{N(p-1)}{N-p}<p<r<p^*$, with $p^*=\frac{Np}{N-p}$. We will guarantee the existence of a solution in the Nehari manifold. Further by using the Lusternik-Schnirelman category we will prove the existence of at least $\text{cat}(Ω)+1$ number of solutions.
