  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 $$\left(-\Delta_p\right)^{\alpha}u +
\left(-\Delta_q\right)^{\beta}u \,= \lambda\left[a(x) \left|u\right|^{p-2}u +
b(x) \left|u\right|^{q-2}u \right]\quad\quad\textrm{in $\O$},$$ where $\O$ is a
smooth bounded domain in $\R^N$ extended by zero outside. When $\O=\R^N$ and
$b\equiv0$, we further show that there exists a continuous family of the
eigenvalue if $1<q<p<q^*_\beta=\frac{Nq}{N-q\beta}$ 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 $\dfrac{p-t}{p_{\alpha}^*}+
\dfrac{p\left(1-t\right)}{s} =1$, 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 $W^{\alpha, p}\left(\R^N\right) \hookrightarrow W^{\beta,
q}\left(\R^N\right)$, we must employ the concentration-compactness principle of
P.L. Lions \cite{PLL} to overcome the difficulty.
