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We study the eigenvalue problem $$ (-\Delta)^s u(x)+ V(x)u(x)-K(x)|u|^{p-2}u(x) =\lambda u(x) \quad \text{in } \mathbb{R}^N, $$ where $s\in(0,1)$, $N>2s$, $2<p<2^{*}=\frac{2N}{N-2s}$, V(x) is indefinite and allowed to be unbounded from below, and K(x) is nonnegative and allowed to be unbounded from above. When $\lambda <\lambda_0=\inf \sigma((-\Delta)^s +V(x))$ (the lowest spectrum of the operator $(-\Delta)^s +V(x))$, we obtain a positive ground state solution by using the constrained minimization method. Also we discuss the regularity of solutions.