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

Abstract

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 $${\left(-{\Delta }_p\right)}^{\alpha }u+{\left(-{\Delta }_q\right)}^{\beta }u=\lambda \left[a(x){\left|u\right|}^{p-2}u+b(x){\left|u\right|}^{q-2}u\right]\text{in Ø},$$ where $Ø$ is a smooth bounded domain in ${\mathbb{R}}^N$ extended by zero outside. When $Ø={\mathbb{R}}^N$ 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 $0\le a\in L^{{\left(\frac{q_{\beta }^{\ast }}{s}\right)}^{\prime }}\left({\mathbb{R}}^N\right)\bigcap L^{\infty }\left({\mathbb{R}}^N\right)$ with $s$ satisfies $\frac{p-t}{p_{\alpha }^{\ast }}+\frac{p\left(1-t\right)}{s}=1$, for some $t\in \left(0,\sqrt{\frac{p-q}{p}}\right).$ 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 $W^{\alpha ,p}\left({\mathbb{R}}^N\right)\hookrightarrow W^{\beta ,q}\left({\mathbb{R}}^N\right)$, we must employ the concentration-compactness principle of P.L. Lions \cite{PLL} to overcome the difficulty.

References in corpus (4)

• Positive solutions for nonlinear Neumann problems with singular terms and convection
• Global bifurcation for fractional $p$-Laplacian and application
• Spectrum of the fractional $p-$Laplacian in ${\mathbb{R}}^N$ and decay estimate for positive solutions of a Schrödinger equation
• Local behaviour and existence of solutions of the fractional (p,q)-Laplacian

Cited by in corpus (1)

• On the existence of three non-negative solutions for a $(p,q)$-Laplacian system