A compact embedding result for nonlocal Sobolev spaces and multiplicity of sign-changing solutions for nonlocal Schrödinger equations

For any s ∈ (0, 1), let the nonlocal Sobolev space X^s(ℝ^N) be the linear space of Lebesgue measure functions from ℝ^N to ℝ such that any function u in X^s(ℝ^N) belongs to L²(ℝ^N) and the function

\((x,y)\longmapsto\big(u(x)-u(y)\big)\sqrt{K(x-y)}\) $$(x,y)⟼(u(x)-u(y))\sqrt{K(x-y)}$$

is in L²(ℝ^N, ℝ^N). First, we show, for a coercive function V(x), the subspace

\(E:=\bigg\{u\in X^s(\mathbb{R}^N):\int_{\mathbb{R}^N}V(x)u^2{\rm d}x<+\infty\bigg\}\) $$E:=\{u\in X^s({\mathbb{R}}^N):{\int }_{{\mathbb{R}}^N}V(x)u^2\mathrm{d}x<+\mathrm{\infty }\}$$

of X^s(ℝ^N) is embedded compactly into L^p(ℝ^N) for \(p\in[2,2_s^*)\) $$p\in [2,2_s^{\ast })$$ , where \(2_s^*\) $$2_s^{\ast }$$ is the fractional Sobolev critical exponent. In terms of applications, the existence of a least energy sign-changing solution and infinitely many sign-changing solutions of the nonlocal Schrödinger equation

\(-{\cal{L}_K}u+V(x)u=f(x,u),\ x\in\ \mathbb{R}^N\) $$-{\mathcal{L}}_{K}u+V(x)u=f(x,u),x\in {\mathbb{R}}^N$$

are obtained, where \(-{\cal{L}_K}\) $$-{\mathcal{L}}_K$$ is an integro-differential operator and V is coercive at infinity.