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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.

In this paper, we are concerned with the global bifurcation results for quasilinear elliptic problem − d i v (φ p (∇ y)) = λ a (x) φ p (y) + a (x) f (x , y , λ) + g (x , y , λ) , i n B , y = 0 , o n ∂ B , where λ is a parameter, f , g ∈ C (B × R × R , R). Let B be a unit open ball of R N with a smooth boundary ∂ B. We shall show that there are two distinct unbounded continua C k + and C k − , consisting of the bifurcation branch C k if f is not necessarily differentiable at the origin with respect to φ p (y) , and there are two distinct unbounded continua D k + and D k − , consisting of the bifurcation branch D k if f is not necessarily differentiable at infinity with respect to φ p (y). As the applications of the above result, we shall prove more details about the existence and multiplicity results of sign-changing solutions for the elliptic problem − d i v (φ p (∇ y)) = λ a (x) f (y) + a (x) g (y) , i n B , y = 0 , o n ∂ B , where f , g ∈ C (R , R) and g is not necessarily differentiable at the origin and infinity with respect to φ p (y). Furthermore, by using a comparison theorem, we also obtain a non-existence result of nodal solutions to the above problem. [ABSTRACT FROM AUTHOR]

We apply the continuation theorem of Mawhin to ensure that a fourth-order nonlinear difference equation of the form Δ 4 u k − 2 − a k u α k + b k u β k = 0 with periodic boundary conditions possesses at least one nontrivial positive solution, where Δ u k = u k + 1 − u k is the forward difference operator, α > 0 , β > 0 and α ≠ β , a k , b k are T -periodic functions and a k b k > 0. As applications, we give some examples to illustrate the application of these theorems. [ABSTRACT FROM AUTHOR]

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]