nature of the problem.

The paper is organized as follows. In Section \[sec:preliminary\] we collect the notation and some preliminary observations. In Section \[sec:compactness\] we discuss the compactness property of the Euler-Lagrange functional associated with the problem under consideration, while Section \[sec:geometry\] is devoted to its geometric structure. In Section \[sec:nablacondition\] we prove the $\nabla$-condition, which is one of the main ingredient of the critical point theorem we employ in order to get our multiplicity result. Finally, in Section \[sec:proofthmain\] we prove Theorem \[thmain\].

Preliminaries {#sec:preliminary}
=============

In this section we give some preliminary results.

Variational setting {#subsec:variational}
-------------------

First of all, we need some notations. In the sequel we endow the space $X_0$ with the norm defined as (see [@svmountain Lemma 6]) $$\label{norma}
\|g\|_{X_0}=\Big(\int_{{\mathbb R}^n\times {\mathbb R}^n} |g(x)-g(y)|^2K(x-y)dx\,dy\Big)^{1/2},$$ which is obviously related to the so-called *Gagliardo norm* $$\label{gagliardonorm}
\|g\|_{H^s(\Omega)}=\|g\|_{L^2(\Omega)}+
\Big(\int_{\Omega\times
\Omega}\frac{\,\,\,|g(x)-g(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\Big)^{1/2}$$ of the usual fractional Sobolev space $H^s(\Omega)$. For further details on the fractional Sobolev spaces we refer to [@adams; @valpal; @molicalibro] and to the references therein.

Problem  has a variational structure: indeed, it is the Euler-Lagrange equation of the functional $\mathcal J_\lambda:X_0\to {\mathbb R}$ defined as $$\mathcal J_\lambda(u)=\frac 1 2 \int_{{\mathbb R}^n\times {\mathbb R}^n}|u(x)-u(y)|^2 K(x-y)\,dx\,dy-\frac \lambda 2 \int_\Omega 