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

Teresa Isernia

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.
• Revised: 2020-01-19.
• Accepted: 2020-01-21.
• Published online: 2020-02-17.