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Abstract

In this paper we study a highly nonlocal problem involving a fractional operator combined with a Kirchhoff-type coefficient. The latter is allowed to vanish at the origin (degenerate case). Precisely, we consider the following nonlocal problem $\{\begin{matrix} - M (∥ u ∥_{X_{0}}^{2}) L_{K} u = f (x, u) {(\int_{Ω} (\int_{0}^{u (x)} f (x, τ) d τ) d x)}^{r} & in Ω, \\ u = 0 & in R^{n} ∖ Ω, \end{matrix}$ where $r ⩾ 0$ , $s \in (0, 1)$ , Ω is an open bounded subset of $R^{n}$ , $n > 2 s$ , with continuous boundary, $f : \overline{Ω} \times R \to R$ and $M : [0, + \infty) \to [0, + \infty)$ are functions verifying suitable conditions and $L_{K}$ is the integrodifferential operator defined as follows $L_{K} u (x) : = \int_{R^{n}} (u (x + y) + u (x - y) - 2 u (x)) K (y) d y, x \in R^{n},$ where the kernel K satisfies some natural conditions.

By working in the fractional Sobolev space $X_{0}$ , which encodes Dirichlet homogeneous boundary conditions, and exploiting the genus theory introduced by Krasnoselskii, we derive the existence of infinitely many weak solutions for this problem.

Keywords

Kirchhoff-type equations fractional Laplacian nonlocal problems variational methods critical point theory Krasnoselskii’s genus

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On a fractional Kirchhoff-type equation via Krasnoselskii’s genus

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