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 $p\& q$ 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.

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