Concentrating solutions of the fractional $(p,q)$-Choquard equation with exponential growth
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| Format: | Preprint |
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2025
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| author | Song, Yueqiang Sun, Xueqi Repovš, Dušan D. |
| author_facet | Song, Yueqiang Sun, Xueqi Repovš, Dušan D. |
| contents | This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-Δ)_{p}^{s}u+\varepsilon^{qs}(-Δ)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=\varepsilon^{μ-N}[|x|^{-μ}*F(u)]f(u) \ \ \mbox{in} \ \ \mathbb{R}^N,$$ where $s\in (0,1),$ $\varepsilon>0$ is a parameter, $2\leq p=\frac{N}{s}<q,$ and $0<μ<N.$ The nonlinear function $f$ has an exponential growth at infinity and the continuous potential function $Z$ satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for $\varepsilon>0$ small enough. In a certain sense, we generalize some previously known results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentrating solutions of the fractional $(p,q)$-Choquard equation with exponential growth Song, Yueqiang Sun, Xueqi Repovš, Dušan D. Analysis of PDEs 35A15, 35A23, 35J35, 35J60, 35R11 This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-Δ)_{p}^{s}u+\varepsilon^{qs}(-Δ)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=\varepsilon^{μ-N}[|x|^{-μ}*F(u)]f(u) \ \ \mbox{in} \ \ \mathbb{R}^N,$$ where $s\in (0,1),$ $\varepsilon>0$ is a parameter, $2\leq p=\frac{N}{s}<q,$ and $0<μ<N.$ The nonlinear function $f$ has an exponential growth at infinity and the continuous potential function $Z$ satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for $\varepsilon>0$ small enough. In a certain sense, we generalize some previously known results. |
| title | Concentrating solutions of the fractional $(p,q)$-Choquard equation with exponential growth |
| topic | Analysis of PDEs 35A15, 35A23, 35J35, 35J60, 35R11 |
| url | https://arxiv.org/abs/2506.00412 |