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Minbo Yang - One of the best experts on this subject based on the ideXlab platform.

  • the brezis nirenberg type critical problem for the nonlinear choquard equation
    Science China-mathematics, 2018
    Co-Authors: Fashun Gao, Minbo Yang
    Abstract:

    We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$ - \Delta u = \left( {\int_\Omega {\frac{{{{\left| {u\left( y \right)} \right|}^{2_\mu ^*}}}}{{{{\left| {x - y} \right|}^\mu }}}dy} } \right){\left| u \right|^{2_\mu ^* - 2}}u + \lambda uin\Omega ,$$ , where Ω is a bounded domain of R N with Lipschitz Boundary, λ is a real parameter, N ≥ 3, $$2_\mu ^* = \left( {2N - \mu } \right)/\left( {N - 2} \right)$$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

  • on the brezis nirenberg type critical problem for nonlinear choquard equation
    arXiv: Analysis of PDEs, 2016
    Co-Authors: Fashun Gao, Minbo Yang
    Abstract:

    We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda u\4.14mm\mbox{in}\1.14mm \Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz Boundary, $\lambda$ is a real parameter, $N\geq3$, $2_{\mu}^{\ast}=(2N-\mu)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

Prange Christophe - One of the best experts on this subject based on the ideXlab platform.

  • Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law
    HAL CCSD, 2019
    Co-Authors: Higaki Mitsuo, Prange Christophe
    Abstract:

    42 pagesWe investigate regularity estimates for the stationary Navier-Stokes equations above a highly oscillating Lipschitz Boundary with the no-slip Boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the Boundary layer thickness. We also obtain an improved $C^{1,\mu}$ estimate and identify the building blocks of the regularity theory, dubbed `Navier polynomials'. In the case when some structure is assumed on the oscillations of the Boundary, for instance periodicity, these estimates can be seen as local error estimates. Although we handle the regularity of the nonlinear stationary Navier-Stokes equations, our results do not require any smallness assumption on the solutions

  • Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law
    2019
    Co-Authors: Higaki Mitsuo, Prange Christophe
    Abstract:

    We investigate regularity estimates for the stationary Navier-Stokes equations above a highly oscillating Lipschitz Boundary with the no-slip Boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the Boundary layer thickness. We also obtain an improved $C^{1,\mu}$ estimate and identify the building blocks of the regularity theory, dubbed `Navier polynomials'. In the case when some structure is assumed on the oscillations of the Boundary, for instance periodicity, these estimates can be seen as local error estimates. Although we handle the regularity of the nonlinear stationary Navier-Stokes equations, our results do not require any smallness assumption on the solutions.Comment: 42 page

Sanghyeon Yu - One of the best experts on this subject based on the ideXlab platform.

Binlin Zhang - One of the best experts on this subject based on the ideXlab platform.

  • fractional kirchhoff problems with critical trudinger moser nonlinearity
    Calculus of Variations and Partial Differential Equations, 2019
    Co-Authors: Xiang Mingqi, Vicenţiu D Rădulescu, Binlin Zhang
    Abstract:

    This paper is concerned with the existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity: $$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy\right) (-\Delta )^{s}_{N/s}u=f(x,u)\,\, \ &{}\quad \mathrm{in}\ \Omega ,\\ u=0\ \ \ \ &{}\quad \mathrm{in}\ {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. } \end{aligned}$$ where $$(-\Delta )^{s}_{N/s}$$ is the fractional N / s-Laplacian operator, $$N\ge 1$$ , $$s\in (0,1)$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz Boundary, $$M:{\mathbb {R}}^+_0\rightarrow {\mathbb {R}}^+_0$$ is a continuous function, and $$f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}} $$ is a continuous function behaving like $$\exp (\alpha t^{2})$$ as $$t\rightarrow \infty $$ for some $$\alpha >0$$ . We first obtain the existence of a ground state solution with positive energy by using minimax techniques combined with the fractional Trudinger–Moser inequality. Next, the existence of nonnegative solutions with negative energy is established by using Ekeland’s variational principle. The main feature of this paper consists in the presence of a (possibly degenerate) Kirchhoff model, combined with a critical Trudinger–Moser nonlinearity.

  • degenerate kirchhoff type hyperbolic problems involving the fractional laplacian
    Journal of Evolution Equations, 2018
    Co-Authors: Ning Pan, Patrizia Pucci, Binlin Zhang
    Abstract:

    In this paper, we are concerned with a wave problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following hyperbolic problem involving the fractional Laplacian $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt} +[u]^{2 (\theta -1)}_{s}(-\Delta )^su=|u|^{p-1}u,\ &{}\text{ in } \Omega \times {\mathbb {R}}^{+}, \\ u(\cdot ,0)=u_0,\quad u_t(\cdot ,0)=u_1,&{} \text{ in } \Omega ,\\ u=0,&{} \text{ in } ({\mathbb {R}}^N {\setminus } \Omega )\times {\mathbb {R}}^{+}_0, \end{array}\right. } \end{aligned}$$ where $$[u]_{s}$$ is the Gagliardo seminorm of u, $$s\in (0,1)$$ , $$\theta \in [1, 2_s^*/2)$$ , with $$2_s^*=2N/(N-2s)$$ , $$p\in (2\theta -1, 2_s^*-1]$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz Boundary $$\partial \Omega $$ , $$(-\Delta )^s$$ is the fractional Laplacian. Under some appropriate assumptions, we obtain the global existence, vacuum isolating and blowup of solutions for the above problem by combining the Galerkin method with potential wells theory. Finally, we investigate the existence of global solutions for the above problem with the critical initial conditions. The significant feature and difficulty of the above problem are that the coefficient of $$(-\Delta )^s$$ can vanish at zero.

  • nonlocal kirchhoff diffusion problems local existence and blow up of solutions
    Nonlinearity, 2018
    Co-Authors: Xiang Mingqi, Vicenţiu D Rădulescu, Binlin Zhang
    Abstract:

    In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem where [u] s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz Boundary, is the fractional Laplacian with , is the initial function, and is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of could be zero at the origin.

  • multiplicity of solutions for a class of quasilinear kirchhoff system involving the fractional p laplacian
    Nonlinearity, 2016
    Co-Authors: Mingqi Xiang, Binlin Zhang, Vicenţiu D Rădulescu
    Abstract:

    In this paper, we investigate the multiplicity of solutions for a p-Kirchhoff system driven by a nonlocal integro-differential operator with zero Dirichlet Boundary data. As a special case, we consider the following fractional p-Kirchhoff system {(∑i=1k[ui]s,pp)θ−1(−Δ)psuj(x)=λj|uj|q−2uj+∑i≠jβij|ui|m|uj|m−2ujin Ω,uj=0in RN\Ω, where , , , , is an open bounded subset of with Lipschitz Boundary , N > ps with , is the fractional p-Laplacian, and for , . When and for all , two distinct solutions are obtained by using the Nehari manifold method. When and for all or and for all , the existence of infinitely many solutions is obtained by applying the symmetric mountain pass theorem. To our best knowledge, our results for the above system are new in the study of Kirchhoff problems.

Vicenţiu D Rădulescu - One of the best experts on this subject based on the ideXlab platform.

  • fractional kirchhoff problems with critical trudinger moser nonlinearity
    Calculus of Variations and Partial Differential Equations, 2019
    Co-Authors: Xiang Mingqi, Vicenţiu D Rădulescu, Binlin Zhang
    Abstract:

    This paper is concerned with the existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity: $$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy\right) (-\Delta )^{s}_{N/s}u=f(x,u)\,\, \ &{}\quad \mathrm{in}\ \Omega ,\\ u=0\ \ \ \ &{}\quad \mathrm{in}\ {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. } \end{aligned}$$ where $$(-\Delta )^{s}_{N/s}$$ is the fractional N / s-Laplacian operator, $$N\ge 1$$ , $$s\in (0,1)$$ , $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz Boundary, $$M:{\mathbb {R}}^+_0\rightarrow {\mathbb {R}}^+_0$$ is a continuous function, and $$f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}} $$ is a continuous function behaving like $$\exp (\alpha t^{2})$$ as $$t\rightarrow \infty $$ for some $$\alpha >0$$ . We first obtain the existence of a ground state solution with positive energy by using minimax techniques combined with the fractional Trudinger–Moser inequality. Next, the existence of nonnegative solutions with negative energy is established by using Ekeland’s variational principle. The main feature of this paper consists in the presence of a (possibly degenerate) Kirchhoff model, combined with a critical Trudinger–Moser nonlinearity.

  • nonlocal kirchhoff diffusion problems local existence and blow up of solutions
    Nonlinearity, 2018
    Co-Authors: Xiang Mingqi, Vicenţiu D Rădulescu, Binlin Zhang
    Abstract:

    In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem where [u] s is the Gagliardo seminorm of u, is a bounded domain with Lipschitz Boundary, is the fractional Laplacian with , is the initial function, and is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of could be zero at the origin.

  • multiplicity of solutions for a class of quasilinear kirchhoff system involving the fractional p laplacian
    Nonlinearity, 2016
    Co-Authors: Mingqi Xiang, Binlin Zhang, Vicenţiu D Rădulescu
    Abstract:

    In this paper, we investigate the multiplicity of solutions for a p-Kirchhoff system driven by a nonlocal integro-differential operator with zero Dirichlet Boundary data. As a special case, we consider the following fractional p-Kirchhoff system {(∑i=1k[ui]s,pp)θ−1(−Δ)psuj(x)=λj|uj|q−2uj+∑i≠jβij|ui|m|uj|m−2ujin Ω,uj=0in RN\Ω, where , , , , is an open bounded subset of with Lipschitz Boundary , N > ps with , is the fractional p-Laplacian, and for , . When and for all , two distinct solutions are obtained by using the Nehari manifold method. When and for all or and for all , the existence of infinitely many solutions is obtained by applying the symmetric mountain pass theorem. To our best knowledge, our results for the above system are new in the study of Kirchhoff problems.