Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators
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| Format: | Preprint |
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2026
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| _version_ | 1866917360952147968 |
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| author | Chen, Shaoxiong Hajaiej, Hichem Yang, Min Yang, Zhipeng |
| author_facet | Chen, Shaoxiong Hajaiej, Hichem Yang, Min Yang, Zhipeng |
| contents | We study the Choquard equation involving mixed local and nonlocal operators $$-Δu+(-Δ)^{s}u+V(x)u=(\frac{1}{|x|^μ}* F(u))f(u)\quad\text{in }\R^{2},$$ where $s\in(0,1)$, $μ\in(0,2)$, $F(t)=\int_{0}^{t} f(τ)\,dτ$, and $f$ has subcritical exponential growth of Trudinger--Moser type. Under suitable assumptions on the potential $V$ and the nonlinearity $f$, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23870 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators Chen, Shaoxiong Hajaiej, Hichem Yang, Min Yang, Zhipeng Analysis of PDEs 35R11, 35J61, 35J20 We study the Choquard equation involving mixed local and nonlocal operators $$-Δu+(-Δ)^{s}u+V(x)u=(\frac{1}{|x|^μ}* F(u))f(u)\quad\text{in }\R^{2},$$ where $s\in(0,1)$, $μ\in(0,2)$, $F(t)=\int_{0}^{t} f(τ)\,dτ$, and $f$ has subcritical exponential growth of Trudinger--Moser type. Under suitable assumptions on the potential $V$ and the nonlinearity $f$, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions. |
| title | Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators |
| topic | Analysis of PDEs 35R11, 35J61, 35J20 |
| url | https://arxiv.org/abs/2603.23870 |