Mixed local-nonlocal equations with critical nonlinearity on $\mathbb{R}^N$: Non-existence, Existence, and Multiplicity of positive solutions
Nirjan Biswas, Souptik Chakraborty, Paramananda Das
TL;DR
The paper investigates mixed local-nonlocal elliptic problems with critical nonlinearities on $\mathbb{R}^N$. It develops a variational framework for the mixed operator, proving nonexistence in the homogeneous setting and existence/multiplicity of positive weak solutions under small nonzero data $f$ for the semilinear ($p=2$) and nonlinear ($p>1$) cases, respectively. For the semilinear problem, two positive solutions are obtained via a local minimization and a Mountain Pass argument, aided by energy estimates built from Aubin–Talenti bubbles under the condition $N<6-4s$. For the nonlinear case, a concentration-compactness principle is established to secure at least one positive solution when $f$ is small, with a detailed discussion of the challenges to extend multiplicity to $p\neq 2$. The results highlight the delicate balance between the mixed local-nonlocal operator and critical nonlinearities, and they provide a baseline for further exploration of variational methods in mixed operators.
Abstract
We study the following critical problem involving the mixed local-nonlocal operator: \begin{equation}\label{main_prob_abstract}\tag{$\mathcal{P}_2$} -Δu+(-Δ)^s u=|u|^{2^*-2}u+f(x)\text{ in }\mathbb{R}^N, \end{equation} where $N \ge 3,\, s\in (0,1),\, 2^*= \frac{2N}{N-2}$, and $f$ is a nontrivial non-negative functional which lies in the dual space of the ambient solution space. For $f \equiv0$, ($\mathcal{P}_2$) does not admit any nontrivial weak solution in $L^2(\mathbb{R}^N)$. This phenomenon stands in contrast to the purely local (for $N>4$) and purely nonlocal (for $N>4s$) cases. On the other hand, when $f \not \equiv 0$, we prove the existence of at least two positive weak solutions to ($\mathcal{P}_2$), provided that the dimension $N$ satisfies certain restrictions and $\|f\|$ is small in the corresponding dual space. Next, we consider the nonlinear analogue to ($\mathcal{P}_2$), namely \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 $p \in (1, \infty), N>p$, $p^*=\frac{Np}{N-p}$, $f$ is a nontrivial non-negative functional in the dual space of the ambient solution space. As in the semilinear case, no nontrivial weak solution exists for ($\mathcal{P}_p$) in $L^p(\mathbb{R}^N)$, when $f \equiv 0$. Moreover, for $f \not \equiv 0$ with sufficiently small $\|f\|$, ($\mathcal{P}_p$) admits a positive weak solution. For the existence, we prove a concentration compactness principle for $-Δ_p+(-Δ_p)^s$.