论文标题
量子场理论中的主要特征值和分数$ p-q $ laplace操作员的积极解决方案
Principal eigenvalue and positive solutions for Fractional $P-Q$ Laplace operator in quantum field theory
论文作者
论文摘要
本文介绍了由非均匀分数$ p \＆q $ laplacian操作员带有不确定权重$$ \ left（-Δ_p\ oright）^αu + + weft（ - \左|当$Ø= \ r^n $和$ b \ equiv0 $时，我们进一步表明，如果$ 1 <q <q <q <q <q^*_β= \ frac {nq} {nq} {n-qβ} $和$ 0 l^{\ left（\ frac {q_β^*} {s} \ right）'} \ left（\ r^n \ right）\ bigcap l^{\ infty} \ left（\ r^n \ right（\ r^n \ right），$ s $ s $ s $ s $ \ dfrac {p \ left（1-t \ right）} {s} = 1 $，对于某些$ t \ in \ left（0，\ sqrt {\ dfrac {\ dfrac {p-q} {p-q} {p}}} \ right）。$我们的方法在各种分析中强烈回答，在这种分析中，山上通过了山上的角色。这项研究的主要困难是如何建立Palais-Smale条件。特别是，由于缺乏空间紧凑性和嵌入$ w^{α，p} \ left（\ r^n \ right）\ hookrightArrow w^{β{β} \ left（\ r^n \ right）$，我们必须利用浓度 - 组合的p.l.狮子\ cite {pll}克服困难。
This article deals with the existence and non-existence of positive solutions for the eigenvalue problem driven by nonhomogeneous fractional $p\& q$ Laplacian operator with indefinite weights $$\left(-Δ_p\right)^αu + \left(-Δ_q\right)^βu \,= λ\left[a(x) \left|u\right|^{p-2}u + b(x) \left|u\right|^{q-2}u \right]\quad\quad\textrm{in $Ø$},$$ where $Ø$ is a smooth bounded domain in $\R^N$ extended by zero outside. When $Ø=\R^N$ and $b\equiv0$, we further show that there exists a continuous family of the eigenvalue if $1<q<p<q^*_β=\frac{Nq}{N-qβ}$ and $0\leq a\in L^{\left(\frac{q_β^*}{s}\right)'}\left(\R^N\right)\bigcap L^{\infty}\left(\R^N\right)$ with $s$ satisfies $\dfrac{p-t}{p_α^*}+ \dfrac{p\left(1-t\right)}{s} =1$, for some $t\in \left(0, \sqrt{\dfrac{p-q}{p}}\right).$ Our approach replies strongly on variational analysis, in which the Mountain pass theorem plays the key role. The main difficulty in this study is that how to establish the Palais-Smale conditions. In particular, in $\R^N$, due to the lack of spatial compactness and the embedding $W^{α, p}\left(\R^N\right) \hookrightarrow W^{β, q}\left(\R^N\right)$, we must employ the concentration-compactness principle of P.L. Lions \cite{PLL} to overcome the difficulty.
