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Ground state solutions of scalar field fractional Schrödinger equations

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Abstract

In this paper, we study the existence of multiple ground state solutions for a class of parametric fractional Schrödinger equations whose simplest prototype is

$$\begin{aligned} (-\Delta )^{s}u+V(x)u=\lambda f(x,u) \quad \hbox { in }\mathbb {R}^{n}, \end{aligned}$$

where \(n>2, (-\Delta )^{s}\) stands for the fractional Laplace operator of order \(s\in (0,1)\), and \(\lambda \) is a positive real parameter. The nonlinear term f is assumed to have a superlinear behaviour at the origin and a sublinear decay at infinity. By using variational methods, we establish the existence of a suitable range of positive eigenvalues for which the problem admits at least two nontrivial solutions in a suitable weighted fractional Sobolev space.

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Acknowledgments

The manuscript was realized within the auspices of the INdAM–GNAMPA Project 2015 titled Modelli ed equazioni non-locali di tipo frazionario. V. Rădulescu acknowledges the support through Grant CNCS PCE-47/2011.

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Authors and Affiliations

  1. Dipartimento PAU, Università ‘Mediterranea’ di Reggio Calabria, Via Graziella, Feo di Vito, 89124, Reggio Calabria, Italy

    Giovanni Molica Bisci

  2. Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Vicenţiu D. Rădulescu

  3. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700, Bucharest, Romania

    Vicenţiu D. Rădulescu

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  1. Giovanni Molica Bisci
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Correspondence to Giovanni Molica Bisci.

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Communicated by P. Rabinowitz.

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Bisci, G.M., Rădulescu, V.D. Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. 54, 2985–3008 (2015). https://doi.org/10.1007/s00526-015-0891-5

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