Nonlocal Schrödinger-Kirchhoff equations with external magnetic field
Mingqi Xiang Patrizia Pucci Marco Squassina Binlin Zhang
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1631-1649 doi: 10.3934/dcds.2017067
The paper deals with the existence and multiplicity of solutions of the fractional Schrödinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case
$\begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \text{in $\mathbb{R}^N$},\end{equation*}$
$s∈ (0,1)$
$a∈ \mathbb{R}^+_0$
$b∈ \mathbb{R}^+_0$
is a magnetic potential,
$V:\mathbb{R}^N\to \mathbb{R}^+$
is an electric potential,
$(-Δ )_A^s$
is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.
keywords: Schrödinger-Kirchhoff equation fractional magnetic operators.
A critical fractional p-Kirchhoff type problem involving discontinuous nonlinearity
Mingqi Xiang Binlin Zhang
Discrete & Continuous Dynamical Systems - S 2019, 12(2): 413-433 doi: 10.3934/dcdss.2019027
The aim of this paper is to discuss the existence and multiplicity of solutions for the following fractional
-Kirchhoff type problem involving the critical Sobolev exponent and discontinuous nonlinearity:
$\begin{align*}M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy\right)(-\Delta)_p^su = \lambda|u|^{p_s^*-2}u+f(x,u)~~\mbox{in }\,\,\mathbb{R}^N,\end{align*}$
$M(t) = a+bt^{\theta-1}$
$t\geq 0$
$a\geq 0, b>0,\theta>1$
is the fractional
--Laplacian with
$p_s^* = Np/(N-ps)$
is the critical Sobolev exponent,
is a parameter, and
is a function. Under suitable assumptions on
, we show that there exists
such that the above equation admits at least one nontrivial nonnegative solution provided
by using the nonsmooth critical point theory for locally Lipschitz functionals. Furthermore, for any
, there exists
such that the above equation has
pairs of nontrivial solutions if
. The main feature is that our paper covers the degenerate case, that is the coefficient of
may be zero at zero. Moreover, the existence results are obtained when
is discontinuous. Thus, our results are new even in the semilinear case.
keywords: Kirchhoff problem fractional p-Laplacian critical Sobolev exponent discontinuous nonlinearity principle of concentration compactness
A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian
Patrizia Pucci Mingqi Xiang Binlin Zhang
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 4035-4051 doi: 10.3934/dcds.2017171

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 $\partialΩ$, $1 < p < N/s$, with $0 < s < 1$, the main Kirchhoff function $M:\mathbb{R}^{ + }_{0} \to \mathbb{R}^{ + }$ is a continuous and nondecreasing function, $(-Δ)_p^s$ is the fractional $p$-Laplacian, $u_0$ is in $L^2(Ω)$ and $f∈ L^2_{\rm loc}(\mathbb{R}^{ + }_0;L^2(Ω))$. Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the large-time behavior and extinction of solutions are also investigated.

keywords: Integro-differential operators anomalous diffusion models sub-differential approach large-time behavior
Existence of weak solutions for non-local fractional problems via Morse theory
Massimiliano Ferrara Giovanni Molica Bisci Binlin Zhang
Discrete & Continuous Dynamical Systems - B 2014, 19(8): 2483-2499 doi: 10.3934/dcdsb.2014.19.2483
In this paper, we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. Non-trivial solutions are obtained by computing the critical groups and Morse theory. Our results extend some classical theorems for semilinear elliptic equations to the non-local fractional setting.
keywords: Fractional Laplacian Morse theory. integro-differential operator critical groups

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