A Harnack type inequality for some conformally invariant equations on half Euclidean space

Existence of infinitely many solutions for a nonlocal elliptic PDE involving singularity

“…Here the authors have used the Lusternik-Schnirelman's theory to guarantee the existence of infinitely many solutions. The problem in [29] was further generalized by Li and Zhang [30] with the driving operator being −∆ p − ∆ q . The reader may also refer to the work due to Figueiredo [31].…”

Existence, multiplicity and concentration for a class of fractional <inline-formula><tex-math id="M1">\begin{document}$ p \&amp; q $\end{document}</tex-math></inline-formula> Laplacian problems in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R} ^{N} $\end{document}</tex-math></inline-formula>

“…Indeed, in these applications, u stands for a concentration, div(D(u)∇u) is the diffusion with diffusion coefficient D(u), and the reaction term c(x, u) relates to source and loss processes. We point out that classical p&q Laplacian problems in bounded or unbounded domains have been studied by several authors; see for instance [17,20,27,28,32,37,38,42] and the references therein. However, the study of the fractional p-Laplacian operator has achieved a tremendous popularity in the last decade.…”