Opuscula Math. 40, no. 1 (2020), 93-110
https://doi.org/10.7494/OpMath.2020.40.1.93
Opuscula Mathematica
Fractional p&q-Laplacian problems with potentials vanishing at infinity
Abstract. 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.
Keywords: fractional \(p\&q\)-Laplacian, vanishing potentials, ground state solution.
Mathematics Subject Classification: 35A15, 35J60, 35R11, 45G05.
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- Teresa Isernia
https://orcid.org/0000-0002-6215-3219
- Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy
- Communicated by Patrizia Pucci.
- Received: 2019-12-15.
- Revised: 2020-01-19.
- Accepted: 2020-01-21.
- Published online: 2020-02-17.