Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities
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...here exist z 0 ∈ X 0 such that zn ⇀ z 0 weakly in X 0 as n → ∞. Let kzn − z 0k → a2 and RR Ω×Ω (zn−z0) 2∗ µ(zn−z0) 2∗ µ |x−y|µ dxdy→ b 22∗ µ as n→ ∞. By the mean value theorem, Brezis-Lieb Lemma (see [7, 17]) and (6.1), we deduce that Z Ω G(x,z 0) dx≥ Z Ω G(x,zn) dx+ Z Ω g(x,zn)(z 0 −zn) dx ≥ Z Ω G(x,zn) dx−λ ZZ Ω×Ω (zn+u)2 ∗ µ(zn+u)2 ∗ µ−1(z n−z 0) |x−y|µ dxdy −hξn,zn−z 0i +hzn,zn−z 0i = Z Ω G(x,zn) dx−...
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"Singular doubly nonlocal elliptic p..." refers background in this paper
...Pekar model of the polaron, where free electrons in an ionic lattice interact with photons associated to the deformations of the lattice or with the polarization that it creates on the medium [15, 16]....
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"Singular doubly nonlocal elliptic p..." refers background in this paper
...malized (kφ 1kL∞(Ω) = 1) eigenfunction corresponding to the smallest eigenvalue of (−∆)son X 0 and A>diam(Ω). We recall that φ 1 ∈ Cs(RN) and φ 1 ∈ C+ ds(Ω) (See Proposition 1.1 and Theorem 1.2 of [40]). Before giving the definition of weak solution to (Pλ) we discuss the solution of the following purely singular problem (−∆)su= u−q,u>0 in Ω,u= 0 in RN\ Ω. (2.1) From [2] we know the following res...
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"Singular doubly nonlocal elliptic p..." refers background or methods in this paper
...From the embedding results ([32]), the space X0 is continuously embedded into Lr(RN ) with r ∈ [1, 2s ] where 2s = 2N N−2s ....
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...2) have been extensively studied in [32, 1] and references therein....
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596 citations
"Singular doubly nonlocal elliptic p..." refers methods in this paper
...ollowing problem (−∆)su= λ a(x) ur +f(x,u) in Ω, u= 0 in RN\ Ω (1.3) where Ωis a bounded domain with smooth boundary, N>2s,0 <s<1,r,λ>0,f(x,u) ∼ up,1 <p<2∗ s−1. In the spirit of [12], here authors first prove the existence of solutions unto the equation with singular term 1/urreplaced by 1/(u+1/n)r and use the uniform estimates on the sequence {un} to finally prove the existence of...
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...an defined as (−∆)su(x) = −P.V. Z RN u(x) −u(y) |x−y|N+2s dy where P.V denotes the Cauchy principal value. The problems involving singular nonlinearity have a very long history. In the pioneering work [12], Crandall, Rabinowitz and Tartar [12] proved the existence of a solution of classical elliptic PDE with singular nonlinearity using the approximation arguments. Later many researchers studied the pro...
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