论文标题
$（p，q）$ - 带有两个参数的laplace方程的积极解决方案的多样性
Multiplicity of positive solutions for $(p,q)$-Laplace equations with two parameters
论文作者
论文摘要
我们研究方程的零dirichlet问题$-Δ_Pu-Δ_Qu =α| u |^{p-2} u+β| | u | U |^{q-2 {q-2} u $在一个有界域$ω\ subset \ subset \ subset \ mathbb {r}^n $中，带有$ 1 <q <q <p $。我们研究了$（α，β）$ - 平面的两个临界曲线之间的关系，该平面与特殊类别溶液的存在的阈值相对应。特别是，在点$（α，β）= \ left的某些社区中（\ | \nablaφ_p\ | _p^p/\ | c/\ |；在$ p $ -laplacian中，我们表明了两个的存在，这是出乎意料的，这是三个独特的积极解决方案，具体取决于指数$ p $和$ q $之间的关系。
We study the zero Dirichlet problem for the equation $-Δ_p u -Δ_q u = α|u|^{p-2}u+β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on the $(α,β)$-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighbourhoods of the point $(α,β) = \left(\|\nabla φ_p\|_p^p/\|φ_p\|_p^p, \|\nabla φ_p\|_q^q/\|φ_p\|_q^q\right)$, where $φ_p$ is the first eigenfunction of the $p$-Laplacian, we show the existence of two and, which is rather unexpected, three distinct positive solutions, depending on a relation between the exponents $p$ and $q$.
