Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion

Borkowski, Dariusz; Jańczak-Borkowska, Katarzyna

doi:10.7494/opmath.2018.38.3.307

Cited by 5 publications

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References 16 publications

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“…This equation has been introduced for the first time by Laskin [35] as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths. Lately, the study of fractional Schrödinger equations has attracted the attention of many mathematicians and several papers appeared studying existence, multiplicity, regularity and asymptotic behavior of solutions to fractional Schrödinger equations assuming different conditions on the potential and considering nonlinearities with subcritical or critical growth; see [7,9,13,21,29,33,34,39,43].…”

Section: Introductionmentioning

confidence: 99%

Fractional p&q-Laplacian problems with potentials vanishing at infinity

Isernia¹

2020

Opuscula Math.

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In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator.

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