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

Abstract

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 $s \in (0,1), p \in (1, \infty), N>p$, $p^*=\frac{Np}{N-p}$, and $f$ is a nonnegative functional in the dual space of the ambient solution space. If $f \equiv0$, 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 $\|f\|$ is small. For this purpose we prove the concentration compactness principle for the mixed operator $-Δ_p +(-Δ_p)^s$ in $\mathbb{R}^N$. We also discuss the multiplicity of positive weak solutions to \eqref{main_prob_abstract_1}.