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In this paper, we are concerned with the fractional Choquard equation on the whole space R N $\mathbb {R}^N$ ( Δ ) s u = 1 | x | N 2 s u p u p 1 $$\begin{equation*} \hspace*{7pc}(-\Delta )^s u={\left(\frac{1}{|x|^{N-2s}}*u^p\right)}u^{p-1} \end{equation*}$$ with 0 < s < 1 $0<s<1$ , N > 2 s $N>2s$ and p R $p\in \mathbb {R}$ . We first prove that the equation does not possess any positive solution for p 1 $p\le 1$ . When p > 1 $p>1$ , we establish a Liouville type theorem saying that if N < 6 s + 4 s ( 1 + p 2 p ) p 1 , $$\begin{equation*} \hspace*{7pc}N<6s+\frac{4s(1+\sqrt {p^2-p})}{p-1}, \end{equation*}$$ then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.  相似文献   

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Multiple periodic solutions for a nonlinear suspension bridge equation   总被引:1,自引:0,他引:1  
We investigate nonlinear oscillations in a fourth-order partialdifferential equation which models a suspension bridge. Previouswork establishes multiple periodic solutions when a parameterexceeds a certain eigenvalue. In this paper, we use Leray-Schauderdegree theory to prove that if the parameter is increased further,beyond a second eigenvalue, then additional solutions are created.  相似文献   

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We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation -Δu=(integral ((|u(y)|~(2*)_μ/|x-y|~μ)dy) from Ω )|μ|~(2*_μ-2_u)+λu in Ω where Ω is a bounded dotain of R~N with Lipschitz boundary, λ is a real parameter, N≥3,2_μ~*=(2 N-μ)/(N-2)is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.  相似文献   

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We prove the existence for short times of analytic solutions to a Vlasov type equation. The corresponding model is one-dimensional but uses a quite singular force term which involves a full derivative in x of the macroscopic density, making the existence of solutions a difficult question.  相似文献   

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We consider the Choquard equation (also known as the stationary Hartree equation or Schrödinger–Newton equation)
?Δu+u=(Iα?|u|p)|u|p?2u.
Here Iα stands for the Riesz potential of order α(0,N), and N?2N+α<1p12. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when α is either close to 0 or close to N.  相似文献   

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Based on the extended homoclinic test technique, we introduce two new ansätz functions to construct multiple periodic-soliton solutions of Kadomtsev-Petviashvili (KP) equation by the Hirota’s bilinear method. Some entirely new periodic-soliton solutions are obtained. The obtained results show that there exist multiple-periodic solitary waves in the different directions for the KP equation, which differ from complexiton. The employed approach is powerful and can be also applied to solve other nonlinear differential equations.  相似文献   

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In this paper, firstly we find the best constant for a convolution problem on the unit circle via a variational method. Then we apply the best constant on a nonlinear rod equation to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Mathematics Subject Classification (2000) 30C70, 37L05, 35Q58, 58E35  相似文献   

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In this work the existence of a global solution for the mixed problem associated to the nonlinear equation
is proved in a Hilbert space framework by using Galerkin method.  相似文献   

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The article investigates the equation
$ {u_t}{\text{ = }}{\left( {u{u_x}} \right)_x}{\text{ + }}\left( {u - {u_0}} \right)\left( {u - {u_1}} \right){\text{,}}\quad \quad {u_1} > {u_0} > 0. $ {u_t}{\text{ = }}{\left( {u{u_x}} \right)_x}{\text{ + }}\left( {u - {u_0}} \right)\left( {u - {u_1}} \right){\text{,}}\quad \quad {u_1} > {u_0} > 0.  相似文献   

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