780 research outputs found

    Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity

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    This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{ \begin{split} (-\Delta)^su &= \lambda |u|^{q-2}u + \left(\int_{\Omega}\frac{|v(y)|^{2^*_\mu}}{|x-y|^\mu}~\mathrm{d}y\right) |u|^{2^*_\mu-2}u\; \text{in}\; \Omega (-\Delta)^sv &= \delta |v|^{q-2}v + \left(\int_{\Om}\frac{|u(y)|^{2^*_\mu}}{|x-y|^\mu}~\mathrm{d}y \right) |v|^{2^*_\mu-2}v \; \text{in}\; \Omega u &=v=0\; \text{in}\; \mb R^n\setminus\Omega, \end{split} \right. \end{equation*} where Ω\Omega is a smooth bounded domain in \mb R^n, n>2sn >2s, s∈(0,1)s \in (0,1), (−Δ)s(-\Delta)^s is the well known fractional Laplacian, μ∈(0,n)\mu \in (0,n), 2μ∗=2n−μn−2s2^*_\mu = \displaystyle\frac{2n-\mu}{n-2s} is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality, 1<q<21<q<2 and λ,δ>0\lambda,\delta >0 are real parameters. We study the fibering maps corresponding to the functional associated with (Pλ,δ)(P_{\lambda,\delta}) and show that minimization over suitable subsets of Nehari manifold renders the existence of atleast two non trivial solutions of (P_{\la,\delta}) for suitable range of \la and δ\delta.Comment: 37 page

    Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity

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    This paper deals with the existence and the asymptotic behavior of non-negative solutions for a class of stationary Kirchhoff problems driven by a fractional integro-differential operator LK\mathcal L_K and involving a critical nonlinearity. The main feature, as well as the main difficulty, of the analysis is the fact that the Kirchhoff function MM can be zero at zero, that is the problem is degenerate. The adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

    On the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity

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    We consider the fractional Schr\"{o}dinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity ε2sM([u]s,Aε2)(−Δ)Aεsu+V(x)u=\varepsilon^{2s}M([u]_{s,A_\varepsilon}^2)(-\Delta)_{A_\varepsilon}^su + V(x)u = ∣u∣2s∗−2u+h(x,∣u∣2)u,|u|^{2_s^\ast-2}u + h(x,|u|^2)u,   x∈RN,\ \ x\in \mathbb{R}^N, where u(x)→0 u(x) \rightarrow 0 as ∣x∣→∞,|x| \rightarrow \infty, and (−Δ)Aεs(-\Delta)_{A_\varepsilon}^s is the fractional magnetic operator with 0<s<10<s<1, 2s∗=2N/(N−2s),2_s^\ast = 2N/(N-2s), M:R0+→R+M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+} is a continuous nondecreasing function, V:RN→R0+,V:\mathbb{R}^N \rightarrow \mathbb{R}^+_0, and A:RN→RNA: \mathbb{R}^N \rightarrow \mathbb{R}^N are the electric and the magnetic potential, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ε<E\varepsilon < \mathcal {E}; and (ii) for any m∗∈Nm^\ast \in \mathbb{N}, has m∗m^\ast pairs of solutions if ε<Em∗\varepsilon < \mathcal {E}_{m^\ast}, where E\mathcal {E} and Em∗\mathcal {E}_{m^\ast} are sufficiently small positive numbers. Moreover, these solutions uε→0u_\varepsilon \rightarrow 0 as ε→0\varepsilon \rightarrow 0
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