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A COMPACT EMBEDDING RESULT FOR NONLOCAL SOBOLEV SPACES AND MULTIPLICITY OF SIGN-CHANGING SOLUTIONS FOR NONLOCAL SCHRÖDINGER EQUATIONS*
Xu Zhang, Hao zhai, Fukun zhao
Acta mathematica scientia,Series B
2024, 44 (5):
1853-1876.
DOI: 10.1007/s10473-024-0512-5
For any $s\in(0,1)$, let the nonlocal Sobolev space $X^s(\mathbb{R} ^N)$ be the linear space of Lebesgue measure functions from $\mathbb{R} ^N$ to $\mathbb{R} $ such that any function $u$ in $X^s(\mathbb{R} ^N)$ belongs to $L^2(\mathbb{R} ^N)$ and the function $(x,y)\longmapsto\big(u(x)-u(y)\big)\sqrt{K(x-y)}$ is in $L^2(\mathbb{R} ^N,\mathbb{R} ^N)$. First, we show, for a coercive function $V(x)$, the subspace $E:=\bigg\{u\in X^s(\mathbb{R} ^N):\int_{\mathbb{R} ^N}V(x)u^2{\rm d}x<+\infty\bigg\}$ of $X^s(\mathbb{R} ^N)$ is embedded compactly into $L^p(\mathbb{R}^N)$ for $p\in[2,2_s^*)$, where $2_s^*$ 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 $-{\mathcal{L}_K}u+V(x)u=f(x,u),\ x\in\ \mathbb{R} ^N$ are obtained, where $-{\mathcal{L}_K}$ is an integro-differential operator and $V$ is coercive at infinity.
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