Abstract:In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators:
\begin{equation*}
\begin{array}{rcl}
-\Delta_p u +(-\Delta_p)^s u & = & \lambda |u|^{p-2}u +\mu |u|^{q-2}u +(I_{\alpha}*|u|^{p^*_{\alpha}})|u|^{p^*_{\alpha}-2}u \text{ in } \mathbb{R}^N;
\left\| u \right\|_p & = & \tau.
\end{array}
\end{equation*}
Here, $N\geq 3$, $2 \le p<N$, $\tau>0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (\max\{0,N-2p\}, N)$, $p^*_{\alpha}=\frac{p}{2}\left(\frac{N+\alpha}{N-p}\right)$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-\Delta_p)^s$ is the non-local fractional p-Laplacian operator with $s\in (0,1)$, $\mu>0$ is a parameter and $\lambda$ appears as a Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of a mass subcritical perturbation, $\mu |u|^{q-2}u$ with $p<q<p+\frac{sp^2}{N}$ under some conditions on $p,N$ and $s$.
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | 35Q55, 35M10, 35J62, 35A01 |
| Cite as: | arXiv:2605.25787 [math.AP] |
| (or arXiv:2605.25787v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25787 arXiv-issued DOI via DataCite (pending registration) |
From: Nidhi Nidhi [view email]
[v1]
Mon, 25 May 2026 12:34:08 UTC (34 KB)
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