Two nonlinear days in Urbino 2017. Electron. J. Diff. Eqns., Conference 25 (2018), pp. 1-13.

Infinitely many small energy solutions for a fractional Kirchhoff equation involving sublinear nonlinearities

Vincenzo Ambrosio

Abstract:
This article is devoted to the study of the following fractional Kirchhoff equation
$$ M\Big(\int\int_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\Big) (-\Delta)^{s}u+V(x)u=f(x,u) \quad \text{in } \mathbb{R}^{N}, $$
where $(-\Delta)^{s}$ is the fractional Laplacian, $M:\mathbb{R}_{+}\to \mathbb{R}_{+}$ is the Kirchhoff term, $V:\mathbb{R}^{N}\to \mathbb{R}$ is a positive continuous potential and f(x, u) is only locally defined for |u| small. By combining a variant of the symmetric Mountain Pass with a Moser iteration argument, we prove the existence of infinitely many weak solutions converging to zero in $L^{\infty}(\mathbb{R}^{N})$-norm.

Published September 15, 2018.
Math Subject Classifications: 47G20, 35R11, 35A15, 58E05.
Key Words: Fractional Kirchhoff equation; sublinear nonlinearity; symmetric mountain pass.

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Vincenzo Ambrosio
Dipartimento di Scienze Pure e Applicate (DiSPeA)
Università degli Studi di Urbino "Carlo Bo"
Piazza della Repubblica, 13
61029 Urbino (Pesaro e Urbino, Italy)
email: vincenzo.ambrosio@uniurb.it

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