Existence results for mixed local and nonlocal elliptic equations involving singularity and nonregular data
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arXiv
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| Natura: | Preprint |
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2024
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| author | Bhowmick, Souvik Ghosh, Sekhar |
| author_facet | Bhowmick, Souvik Ghosh, Sekhar |
| contents | In this paper, we prove the existence of weak, veryweak and duality solutions to a class of elliptic problems involving singularity and measure data which is given by: $-Δu+(-Δ)^s u = \frac{f(x)}{u^γ} +μ$ in $Ω$ with the zero Dirichlet boundary data $u=0$ in $\mathbb R^N \setminus Ω$. The existence of weak solutions is obtained by approximating a sequence of problems for $0<γ\leq1$ and $γ>1$. We employ Schauder's fixed point theorem and embeddings of Marcinkiewicz spaces. The novelty of our work is that we prove the existence of a duality solution and its equivalence with weak solutions to the problem $\mathcal{L}u=μ$. Moreover, we prove a veryweak maximum principle and a Kato-type inequality for the mixed local-nonlocal operator $\mathcal{L}=-Δ+(-Δ)^s$, which are crucial tools to guarantee the existence of veryweak solutions to the problem. Using a Kato-type inequality, maximum principle together with sub-super solution method, we prove the existence of veryweak solution for $0<γ<1$. Our work extends the studies due to Oliva and Petitta [ESAIM Control Optim. Calc. Var., 22(1):289--308, 2016.] and Petitta [Adv. Nonlinear Stud., 16(1):115--124, 2016.] for the mixed local-nonlocal operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04441 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence results for mixed local and nonlocal elliptic equations involving singularity and nonregular data Bhowmick, Souvik Ghosh, Sekhar Analysis of PDEs 35M12, 35J75, 35R06, 35R11, 35J20 In this paper, we prove the existence of weak, veryweak and duality solutions to a class of elliptic problems involving singularity and measure data which is given by: $-Δu+(-Δ)^s u = \frac{f(x)}{u^γ} +μ$ in $Ω$ with the zero Dirichlet boundary data $u=0$ in $\mathbb R^N \setminus Ω$. The existence of weak solutions is obtained by approximating a sequence of problems for $0<γ\leq1$ and $γ>1$. We employ Schauder's fixed point theorem and embeddings of Marcinkiewicz spaces. The novelty of our work is that we prove the existence of a duality solution and its equivalence with weak solutions to the problem $\mathcal{L}u=μ$. Moreover, we prove a veryweak maximum principle and a Kato-type inequality for the mixed local-nonlocal operator $\mathcal{L}=-Δ+(-Δ)^s$, which are crucial tools to guarantee the existence of veryweak solutions to the problem. Using a Kato-type inequality, maximum principle together with sub-super solution method, we prove the existence of veryweak solution for $0<γ<1$. Our work extends the studies due to Oliva and Petitta [ESAIM Control Optim. Calc. Var., 22(1):289--308, 2016.] and Petitta [Adv. Nonlinear Stud., 16(1):115--124, 2016.] for the mixed local-nonlocal operator. |
| title | Existence results for mixed local and nonlocal elliptic equations involving singularity and nonregular data |
| topic | Analysis of PDEs 35M12, 35J75, 35R06, 35R11, 35J20 |
| url | https://arxiv.org/abs/2410.04441 |