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We consider a class of generalized nonlocal $p$-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.
In this paper, we establish a new fractional interpolation inequality for radially symmetric measurable functions on the whole space $R^{N}$ and a new compact imbedding result about radially symmetric measurable functions. We show that the best constant in the new interpolation inequality can be achieved by a radially symmetric function. As applications of this compactness result, we study the existence of ground states of the nonlinear fractional Schrödinger equation on the whole space $R^{N}$. We also prove an existence result of standing waves and prove their orbital stability.
Experimental results on the immune response to cancer indicate that activation of cytotoxic T lymphocytes (CTLs) through interactions with dendritic cells (DCs) can trigger a change in CTL migration patterns. In particular, while CTLs in the pre-activation state move in a non-local search pattern, the search pattern of activated CTLs is more localised. In this paper, we develop a kinetic model for such a switch in CTL migration modes. The model is formulated as a coupled system of balance equations for the one-particle distribution functions of CTLs in the pre-activation state, activated CTLs and DCs. CTL activation is modelled via binary interactions between CTLs in the pre-activation state and DCs. Moreover, cell motion is represented as a velocity-jump process, with the running time of CTLs in the pre-activation state following a long-tailed distribution, which is consistent with a Lévy walk, and the running time of activated CTLs following a Poisson distribution, which corresponds to Brownian motion. We formally show that the macroscopic limit of the model comprises a coupled system of balance equations for the cell densities, whereby activated CTL movement is described via a classical diffusion term, whilst a fractional diffusion term describes the movement of CTLs in the pre-activation state. The modelling approach presented here and its possible generalisations are expected to find applications in the study of the immune response to cancer and in other biological contexts in which switch from non-local to localised migration patterns occurs.
where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$, $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$, respectively.
In this paper, we concern with a backward problem for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence and uniqueness of the mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution and further we prove the convergence rate for the regularized solutions.
Dye-sensitized solar cells consistently provide a cost-effective avenue for sources of renewable energy, primarily due to their unique utilization of nanoporous semiconductors. Through mathematical modelling, we are able to uncover insights into electron transport to optimize the operating efficiency of the dye-sensitized solar cells. In particular, fractional diffusion equations create a link between electron density and porosity of the nanoporous semiconductors. We numerically solve a fractional diffusion model using a finite-difference method and a finite-element method to discretize space and an implicit finite-difference method to discretize time. Finally, we calculate the accuracy of each method by evaluating the numerical errors under grid refinement.
In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.
In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations
\begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*}
where $0<\alpha <1$, $\Omega = \mathbb {R}^{N}$ or $\Omega$ is a smooth bounded domain, $\Gamma$ is a singular subset of $\Omega$ with fractional capacity zero, $f(t)$ is locally bounded and positive for $t\in [0,\,\infty )$, and $f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in $t$ for large $t$, rather than for every $t>0$. Our main result is that the solutions satisfy the estimate
\begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*}
This estimate is new even for $\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.
We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.
We consider the fractional elliptic problem:
where B1 is the unit ball in ℝN, N ⩾ 3, s ∈ (0, 1) and p > (N + 2s)/(N − 2s). We prove that this problem has infinitely many solutions with slow decay O(|x|−2s/(p−1)) at infinity. In addition, for each s ∈ (0, 1) there exists Ps > (N + 2s)/(N − 2s), for any (N + 2s)/(N − 2s) < p < Ps, the above problem has a solution with fast decay O(|x|2s−N). This result is the extension of the work by Dávila, del Pino, Musso and Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) to the fractional case.
We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator and a general advection field in of bounded mean oscillation, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by Silvestre, our advection field does not need to be bounded. A similar result can be obtained in the supercritical case if the advection field is Hölder continuous. Our proof is inspired by Kiselev and Nazarov and is based on the dual evolution technique. The idea is to propagate an atom property (i.e., localisation and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.
This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents:
\[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \]
where N ⩾ 4, 0 < s < 1, (y′, y″) ∈ ℝ2 × ℝN−2, ε > 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order $\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$.
In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form − Δ + ( − Δ)s, with s ∈ (0, 1). We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.
In this paper we give sufficient conditions to obtain continuity results of solutions for the so called ϕ-Laplacian Δϕ with respect to domain perturbations. We point out that this kind of results can be extended to a more general class of operators including, for instance, nonlocal nonstandard growth type operators.
where ɛ is a small positive parameter, a and b are positive constants, s ∈ (0, 1) and p ∈ (1, ∞) are such that $sp \in (\frac {3}{2}, 3)$, $(-\Delta )^{s}_{p}$ is the fractional p-Laplacian operator, f: ℝ → ℝ is a superlinear continuous function with subcritical growth and V: ℝ3 → ℝ is a continuous potential having a local minimum. We also prove a multiplicity result and relate the number of positive solutions with the topology of the set where the potential V attains its minimum values. Finally, we obtain an existence result when f(u) = uq−1 + γur−1, where γ > 0 is sufficiently small, and the powers q and r satisfy 2p < q < p*s ⩽ r. The main results are obtained by using some appropriate variational arguments.
Let
$\Omega \subset \mathbb {R}^N$
,
$N\geq 2$
, be an open bounded connected set. We consider the fractional weighted eigenvalue problem
$(-\Delta )^s u =\lambda \rho u$
in
$\Omega $
with homogeneous Dirichlet boundary condition, where
$(-\Delta )^s$
,
$s\in (0,1)$
, is the fractional Laplacian operator,
$\lambda \in \mathbb {R}$
and
$ \rho \in L^\infty (\Omega )$
.
We study weak* continuity, convexity and Gâteaux differentiability of the map
$\rho \mapsto 1/\lambda _1(\rho )$
, where
$\lambda _1(\rho )$
is the first positive eigenvalue. Moreover, denoting by
$\mathcal {G}(\rho _0)$
the class of rearrangements of
$\rho _0$
, we prove the existence of a minimizer of
$\lambda _1(\rho )$
when
$\rho $
varies on
$\mathcal {G}(\rho _0)$
. Finally, we show that, if
$\Omega $
is Steiner symmetric, then every minimizer shares the same symmetry.
where 0 < γ < 1, λ > 0 and 0 < s ≤ t < 1 with 4s + 2t > 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.
where σD is a modulus of continuity defined in the interval [0, |D|] and depends only on the domain D. The inequality is optimal for any domain D and for any values of $\|f\|_1$ and $\|f\|_\infty .$ We also show that
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],
where B is a ball and |B| = |D|. Using this optimality property of σD, we derive Brezis–Galloute–Wainger type inequalities on the L∞ norm of u in terms of the L1 and L∞ norms of f. As an application we derive L∞ − L1 estimates on the k-th Laplace eigenfunction of the domain D.
where n ⩾ 2, 0 < α, β < 2, a> −α, b > −β and p, q ⩾ 1. By exploiting a direct method of scaling spheres for fractional systems, we prove that if $p \leqslant \frac {n+\alpha +2a}{n-\beta }$, $q \leqslant \frac {n+\beta +2b}{n-\alpha }$, $p+q<\frac {n+\alpha +2a}{n-\beta }+\frac {n+\beta +2b}{n-\alpha }$ and (u, v) is a nonnegative strong solution of the system, then u ≡ v ≡ 0.
This paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type
$$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$
where ε > 0 is a parameter, s, t ∈ (0, 1) are such that 2s+2t>3, A:ℝ3 → ℝ3 is a smooth magnetic potential, (−Δ)As is the fractional magnetic Laplacian, V:ℝ3 → ℝ is a continuous electric potential and f:ℝ → ℝ is a C1 subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for e > 0 small enough.