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\\star |\\Psi|^p \\right)|\\Psi|^{p-2}\\Psi\\ \\ \\ \\mathrm{in}\\ \\mathbb{R}^{N+1}, \\] where $N\\geq 2$, $\\alpha\\in (0,1)$, $\\beta\\in (0, N)$, $s\\in (2, 2+\\frac{4\\alpha}{N})$, $p\\in [2, 1+\\frac{2\\alpha+\\beta}{N})$, and the constants $a, \\lambda$ are nonnegative satisfying $a+\\lambda > 0.$ We then extend the 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