Mixed local-nonlocal equations with critical nonlinearity on ${\mathbb{R}}^N$: Non-existence, Existence, and Multiplicity of positive solutions

We consider the following quasilinear critical problem involving the mixed local-nonlocal operator: \begin{equation}\label{main_prob_abstract_1}\tag{${\mathcal{P}}_p$} -Δ_p u+(-Δ_p)^s u=|u|^{p^*-2}u+f(x)$\text{ in }$$\mathbb{R}$^N, \end{equation} where $s\in (0,1),p\in (1,\mathrm{\infty }),N>p$, $p^{\ast }=\frac{Np}{N-p}$, and $f$ is a nonnegative functional in the dual space of the ambient solution space. If $f\equiv 0$, then we show that \eqref{main_prob_abstract_1} does not admit any nontrivial weak solution. This phenomenon stands in contrast to the purely local and purely nonlocal cases. On the other hand, if $f$ is a nontrivial nonnegative functional, we establish the existence of a positive weak solution to \eqref{main_prob_abstract_1} provided $\mathrm{\parallel }f\mathrm{\parallel }$ is small. For this purpose we prove the concentration compactness principle for the mixed operator $-{\mathrm{\Delta }}_p+(-{\mathrm{\Delta }}_p{)}^s$ in ${\mathbb{R}}^N$. We also discuss the multiplicity of positive weak solutions to \eqref{main_prob_abstract_1}.