Principal eigenvalue and positive solutions for Fractional $P-Q$ Laplace operator in quantum field theory

This article deals with the existence and non-existence of positive solutions for the eigenvalue problem driven by nonhomogeneous fractional $p\mathrm{\&}q$ Laplacian operator with indefinite weights $${(-{\mathrm{\Delta }}_p)}^{\alpha }u+{(-{\mathrm{\Delta }}_q)}^{\beta }u=\lambda [a(x){\mid u\mid }^{p-2}u+b(x){\mid u\mid }^{q-2}u]\text{in }\text{O},$$ where $\text{O}$ is a smooth bounded domain in ${\mathbb{R}}^N$ extended by zero outside. When $\text{O}={\mathbb{R}}^N$ and $b\equiv 0$, we further show that there exists a continuous family of the eigenvalue if $1<q<p<q_{\beta }^{\ast }=\frac{Nq}{N-q\beta }$ and $0\leq a\in L^{\left(\frac{q_{\beta}^}{s}\right)'}\left(\R^N\right)\bigcap L^{\infty}\left(\R^N\right)$with$s$satisfies$\dfrac{p-t}{p_{\alpha}^}+ \dfrac{p\left(1-t\right)}{s} =1$,forsome$t\in \left(0, \sqrt{\dfrac{p-q}{p}}\right).$ 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 ${\mathbb{R}}^N$, due to the lack of spatial compactness and the embedding $W^{\alpha, p}\left(\R^N\right) \hookrightarrow W^{\beta, q}\left(\R^N\right)$, we must employ the concentration-compactness principle of P.L. Lions \cite{PLL} to overcome the difficulty.