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This article is devoted to studying the existence of positive solutions to the following fractional Choquard equation: (−Δ)su+u=∫Ω∣u(y)∣p∣x−y∣N−αdy∣u∣p−2u+ε∫Ω∣u(y)∣2α,s*∣x−y∣N−αdy∣u∣2α,s*−2u,inΩ,u=0,onRN\Ω,\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+u=\left(\mathop{\displaystyle \int }\limits_{\Omega }\frac{{| u(y)| }^{p}}{{| x-y| }^{N-\alpha }}{\rm{d}}y\right){| u| }^{p-2}u+\varepsilon \left(\mathop{\displaystyle \int }\limits_{\Omega }\frac{{| u(y)| }^{{2}_{\alpha ,s}^{* }}}{{| x-y| }^{N-\alpha }}{\rm{d}}y\right){| u| }^{{2}_{\alpha ,s}^{* }-2}u,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω\Omega is an exterior domain with smooth boundary ∂Ω≠∅\partial \Omega \ne \varnothing such that RN\Ω{{\mathbb{R}}}^{N}\backslash \Omega is bounded, N>2s,20\varepsilon \gt 0 is a parameter. We establish the limit profiles and uniqueness of positive radial ground-states for the limit equation without the critical exponent as α\alpha sufficiently close to NN. Then, combining variational method, barycentric functions, and Brouwer degree theory, we determine the existence of positive bound-state solutions provided that ε>0\varepsilon \gt 0 is sufficiently small.

We study the fourth-order boundary value problem with a sign-changing weight function: u⁗ = λm(t)u + ƒ1(t, u, u′, u″, u‴, λ) + ƒ2(t, u, u′, u″, u‴, λ), t ∈ (0, 1) , u(0) = u(1) = u″(0) = u″(1) = 0, where λ ∈ ℝ is a parameter, ƒ1, ƒ2 ∈ C([0, 1] × ℝ5, ℝ), ƒ1 is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively. [ABSTRACT FROM AUTHOR]