Showing papers by "Binlin Zhang published in 2018"

Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions

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.

Combined effects for fractional Schrodinger-Kirchhoff systems with critical nonlinearities

Abstract: In this paper, we investigate the existence of solutions for critical Schrodinger–Kirchhoff type systems driven by nonlocal integro–differential operators. As a particular case, we consider the following system: where (–Δ )s p is the fractional p –Laplace operator with 0 /s , α , β > 1 with α + β =p* s , M : ℝ+ 0 → ℝ+ 0 is a continuous function, V : ℝN → ℝ+ is a continuous function, λ > 0 is a real parameter. By applying the mountain pass theorem and Ekeland’s variational principle, we obtain the existence and asymptotic behaviour of solutions for the above systems under some suitable assumptions. A distinguished feature of this paper is that the above systems are degenerate, that is, the Kirchhoff function could vanish at zero. To the best of our knowledge, this is the first time to exploit the existence of solutions for fractional Schrodinger–Kirchhoff systems involving critical nonlinearities in ℝN .

p-fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

Abstract: Abstract This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p-Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.

Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian

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.

A critical Kirchhoff type problem involving the fractional Laplacian in

Abstract: In this paper, we are concerned with the existence of solutions for a critical p-Kirchhoff type problem driven by a nonlocal integro-differential operator:where is a continuous function, is a singular kernel function, is a nonlocal fractional operator, with , , f is a Caratheodory function on satisfying the Ambrosetti–Rabinowitz type condition. Under some suitable assumptions, we obtain the existence of nontrivial solutions for above problem by applying the mountain pass theorem. A distinguished feature of this paper is that M(0) may be zero, which means that the problem is degenerate. Consequently, the main theorem extends in several directions the recent results of Autuori, Fiscella and Pucci [Nonlinear Anal. 2015;125:699–714].