Equations with \(s\)-fractional \((p,q)\)-Laplacian and convolution. (English) Zbl 1487.35467
Summary: This paper deals with a Dirichlet problem on a bounded subdomain \(\Omega \subset \mathbb{R}^N\) for an equation which is doubly nonlocal: it is driven by the (negative) \(s\)-fractional \((p, q)\)-Laplacian for \(s \in (0,1)\) and \(1<q<p<\infty\) and has as reaction term a nonlinearity with an incorporated convolution. Such a problem is considered for the first time. Another major feature concerns the correct formulation for the notion of \(s\)-fractional \((p, q)\)-Laplacian. The stated problem is studied through two different approaches: limit process via finite dimensional approximations and sub-supersolution in the nonlocal setting.
MSC:
| 35S15 | Boundary value problems for PDEs with pseudodifferential operators |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35R11 | Fractional partial differential equations |
| 47G20 | Integro-differential operators |
Keywords:
nonlocal Dirichlet problem; weak solution; \(s\)-fractional \((p, q)\)-Laplacian; convolution; finite dimensional approximation; sub-supersolutionReferences:
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