Quasilinear Laplace equations and inequalities with fractional orders. (English) Zbl 1532.35489
Summary: For the fractional Sobolev space \(H^{s,p}\) on \({\mathbb{R}}^n\) of fundamental interest in geometric potential analysis and partial differential equations, this paper presents a much more subtle dual-capacitary-constructive approach to achieve weak solutions \(u\in H^{s,p}\) of not only the fractional \({{(p,+)}}\)-Laplacian and the fractional \({(p,-)}\)-Laplacian equations (cf. Theorems 1.2–1.3)
\[
\pm \text{div}^s_{\pm }\big (|\nabla^s_{\pm } u|^{p-2}\nabla_{\pm }^su\big ) =\text{ either } f\in L^\frac{pn}{(p-1)n+sp} \text{ or } \delta_0 \text{ -- the Dirac mass at the origin}
\]
but also the fractional \({{(p,+)}} \)-Laplacian and the fractional \({{(p,-)}}\)-Laplacian inequalities (cf. Theorems 1.4–1.5)
\[
\pm \text{div}^s_{\pm }\big (|\nabla^s_{\pm } u|^{p-2}\nabla_{\pm }^s u\big )\ge u^q\ge 0.
\]
MSC:
| 35R11 | Fractional partial differential equations |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35R45 | Partial differential inequalities and systems of partial differential inequalities |
| 42B30 | \(H^p\)-spaces |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 58E35 | Variational inequalities (global problems) in infinite-dimensional spaces |
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