On the multiplicity of weak solutions for a class of coupled quasilinear elliptic systems
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arXiv
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| Format: | Preprint |
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2025
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| author | Canino, Annamaria Mauro, Simone |
| author_facet | Canino, Annamaria Mauro, Simone |
| contents | We study the existence and regularity of weak solutions to the following quasilinear elliptic system: \[
-\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad \text{in } Ω,\quad
u_k = 0 \quad \text{on } \partialΩ, \]
where $k=1,\dots,d$, $ Ω\subset \mathbb{R}^N $ is a bounded domain with $ N \geq 2 $, $ \boldsymbol{p} = (p_1, \dots, p_d) $, $ p_k > 1 $. Using tools from nonsmooth critical point theory, we prove the existence of infinitely many weak solutions in $ W_0^{1,\boldsymbol{p}}(Ω) \cap L^\infty(Ω; \mathbb{R}^d) $, where $W_0^{1,\boldsymbol p}(Ω)=W_0^{1,p_1}(Ω)\times\dots\times W_0^{1,p_d}(Ω)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_22665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the multiplicity of weak solutions for a class of coupled quasilinear elliptic systems Canino, Annamaria Mauro, Simone Analysis of PDEs 35A01, 35A15, 35J05, 35J20, 35J25 We study the existence and regularity of weak solutions to the following quasilinear elliptic system: \[ -\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad \text{in } Ω,\quad u_k = 0 \quad \text{on } \partialΩ, \] where $k=1,\dots,d$, $ Ω\subset \mathbb{R}^N $ is a bounded domain with $ N \geq 2 $, $ \boldsymbol{p} = (p_1, \dots, p_d) $, $ p_k > 1 $. Using tools from nonsmooth critical point theory, we prove the existence of infinitely many weak solutions in $ W_0^{1,\boldsymbol{p}}(Ω) \cap L^\infty(Ω; \mathbb{R}^d) $, where $W_0^{1,\boldsymbol p}(Ω)=W_0^{1,p_1}(Ω)\times\dots\times W_0^{1,p_d}(Ω)$. |
| title | On the multiplicity of weak solutions for a class of coupled quasilinear elliptic systems |
| topic | Analysis of PDEs 35A01, 35A15, 35J05, 35J20, 35J25 |
| url | https://arxiv.org/abs/2511.22665 |