Nonlocal Sublinear Elliptic Problems Involving Measures
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911024905453568 |
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| author | May, Aye Chan Seesanea, Adisak |
| author_facet | May, Aye Chan Seesanea, Adisak |
| contents | We study Dirichlet problems for fractional Laplace equations of the form $(-Δ)^{\fracα{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<α<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} σ_{i} u^{q_i} + ω$ involves sublinear terms with $0<q_{i}<1$ and the coefficients $σ_{i}, ω$ are nonnegative locally finite Borel measures on $\mathbb{R}^n$. We develop a potential theoretic approach for the existence of positive minimal solutions in Lorentz spaces to the problems under certain assumptions on $σ_{i}$ and $ω$. The uniqueness properties of such solutions are discussed. Our techniques are also applicable to similar sublinear problems on uniform bounded domains when $0<α< 2$, or on arbitrary domains with positive Green's functions in the classical case $α=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12576 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonlocal Sublinear Elliptic Problems Involving Measures May, Aye Chan Seesanea, Adisak Analysis of PDEs 35J61, 31B10, 42B37 We study Dirichlet problems for fractional Laplace equations of the form $(-Δ)^{\fracα{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<α<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} σ_{i} u^{q_i} + ω$ involves sublinear terms with $0<q_{i}<1$ and the coefficients $σ_{i}, ω$ are nonnegative locally finite Borel measures on $\mathbb{R}^n$. We develop a potential theoretic approach for the existence of positive minimal solutions in Lorentz spaces to the problems under certain assumptions on $σ_{i}$ and $ω$. The uniqueness properties of such solutions are discussed. Our techniques are also applicable to similar sublinear problems on uniform bounded domains when $0<α< 2$, or on arbitrary domains with positive Green's functions in the classical case $α=2$. |
| title | Nonlocal Sublinear Elliptic Problems Involving Measures |
| topic | Analysis of PDEs 35J61, 31B10, 42B37 |
| url | https://arxiv.org/abs/2310.12576 |