Showing papers by "Binlin Zhang published in 2017"

A Nonhomogeneous Fractional p-Kirchhoff Type Problem Involving Critical Exponent in ℝ N

Abstract: Abstract This paper concerns itself with the nonexistence and multiplicity of solutions for the following fractional Kirchhoff-type problem involving the critical Sobolev exponent: [ a + b ⁢ ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) θ - 1 ] ⁢ ( - Δ ) p s ⁢ u = | u | p s * - 2 ⁢ u + λ ⁢ f ⁢ ( x ) in ⁢ ℝ N , \\Biggl{[}a+b\\biggl{(}\\iint_{\\mathbb{R}^{2N}}\\frac{\\lvert u(x)-u(y)\\rvert^{p}}{% \\lvert x-y\\rvert^{N+ps}}\\,dx\\,dy\\biggr{)}^{\\theta-1}\\Biggr{]}(-\\Delta)_{p}^{s}% u=\\lvert u\\rvert^{p_{s}^{*}-2}u+\\lambda f(x)\\quad\\text{in }\\mathbb{R}^{N}, where a ≥ 0 {a\\kern-1.0pt\\geq\\kern-1.0pt0} , b > 0 , θ > 1 {b\\kern-1.0pt>\\kern-1.0pt0,\\theta\\kern-1.0pt>\\kern-1.0pt1} , ( - Δ ) p s {(-\\Delta)_{p}^{s}} is the fractional p-Laplacian with 0 < s < 1 {0\\kern-1.0pt<\\kern-1.0pts\\kern-1.0pt<\\kern-1.0pt1} and 1 < p < N / s {1\\kern-1.0pt<\\kern-1.0ptp\\kern-1.0pt<\\kern-1.0ptN/s} , p s * = N ⁢ p / ( N - p ⁢ s ) {p_{s}^{*}\\kern-1.0pt=\\kern-1.0ptNp/(N-ps)} is the critical Sobolev exponent, λ ≥ 0 {\\lambda\\geq 0} is a parameter, and f ∈ L p s * / ( p s * - 1 ) ⁢ ( ℝ N ) ∖ { 0 } {f\\in L^{p_{s}^{*}/(p_{s}^{*}-1)}(\\mathbb{R}^{N})\\setminus\\{0\\}} is a nonnegative function. When λ = 0 {\\lambda=0} , we show that the multiplicity and nonexistence of solutions for the above problem are related with N, θ, s, p, a, and b. When λ > 0 {\\lambda>0} , by using Ekeland’s variational principle and the mountain pass theorem, we show that there exists λ * * > 0 {\\lambda^{**}>0} such that the above problem admits at least two nonnegative solutions for all λ ∈ ( 0 , λ * * ) {\\lambda\\in(0,\\lambda^{**})} . In the latter case, in order to overcome the loss of compactness, we derive a fractional version of the principle of concentration compactness in the setting of the fractional p-Laplacian.

Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian

Abstract: In this paper we study the existence of a global solution for a diffusion problem of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we consider the following parabolic equation involving the fractional p -Laplacian: { ∂ t u + [ u ] s , p ( λ − 1 ) p ( − Δ ) p s u = | u | q − 2 u , in Ω × R + , ∂ t u = ∂ u / ∂ t , u ( x , 0 ) = u 0 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where [ u ] s , p is the Gagliardo p –seminorm of u , Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω , p q N p / ( N − s p ) with 1 p N / s and s ∈ ( 0 , 1 ) , 1 ≤ λ N / ( N − s p ) , ( − Δ ) p s is the fractional p -Laplacian. Under some appropriate assumptions, we obtain the existence of a global solution for the problem above by the Galerkin method and potential well theory. It is worth pointing out that the main result covers the degenerate case, that is the coefficient of ( − Δ ) p s can vanish at zero.

On the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity

Abstract: In this paper, we consider the fractional Schrodinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity e 2 s M ( [ u ] s , A e 2 ) ( − Δ ) A e s u + V ( x ) u = | u | 2 s ∗ − 2 u + h ( x , | u | 2 ) u , x ∈ R N , u ( x ) → 0 , as | x | → ∞ , where ( − Δ ) A e s is the fractional magnetic operator with 0 s 1 , 2 s ∗ = 2 N ∕ ( N − 2 s ) , M : R 0 + → R + is a continuous nondecreasing function, V : R N → R 0 + and A : R N → R N are the electric and magnetic potentials, 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 e E ; and (ii) for any m ∗ ∈ N , has m ∗ pairs of solutions if e E m ∗ , where E and E m ∗ are sufficiently small positive numbers. Moreover, these solutions u e → 0 as e → 0 .

A multiplicity result for asymptotically linear Kirchhoff equations

Abstract: Abstract In this paper, we study the following Kirchhoff type equation: - ( 1 + b ∫ ℝ N | ∇ u | 2 d x ) Δ u + u = a ( x ) f ( u ) in ℝ N , u ∈ H 1 ( ℝ N ) , -\\bigg{(}1+b\\int_{\\mathbb{R}^{N}}\\lvert\ abla u|^{2}\\,dx\\biggr{)}\\Delta u+u=a(% x)f(u)\\quad\\text{in }\\mathbb{R}^{N},\\qquad u\\in H^{1}(\\mathbb{R}^{N}), where N ≥ 3 {N\\geq 3} , b > 0 {b>0} and f ⁢ ( s ) {f(s)} is asymptotically linear at infinity, that is, f ⁢ ( s ) ∼ O ⁢ ( s ) {f(s)\\sim O(s)} as s → + ∞ {s\\rightarrow+\\infty} . By using variational methods, we obtain the existence of a mountain pass type solution and a ground state solution under appropriate assumptions on a ⁢ ( x ) {a(x)} .

A diffusion problem of Kirchhoff type involving the nonlocal fractional p -Laplacian

Abstract: In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1

Ground state solutions and multiple solutions for a p(x)-Laplacian equation in with periodic data

Abstract: In this work, we consider the following p(x)-Laplacian equation in where f, V and p(x) are periodic in . Under some appropriate assumptions, we prove the existence of the ground state solutions via the generalized Nehari method due to Szulkin and Weth. Moreover, if f is odd in u, infinitely many pairs of geometrically distinct solutions are given. To the best of our knowledge, our results are new even in the constant exponent case.