In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in(1/2,1)$, singular nonlinearity, and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.
We study the problem of global exponential stabilization of original Burgers' equations and the Burgers' equation with nonlocal nonlinearities by controllers depending on finitely many parameters. It is shown that solutions of the controlled equations are steering a concrete solution of the non-controlled system as $t\rightarrow \infty$ with an exponential rate.
The aim of this paper is to provide the first application of Theorem 3 of [2] in a case where the dependence of the underlying equation from the real parameter is not of affine type.
We consider a class of scalar field equations with anisotropic nonlocal nonlinearities. We obtain a suitable extension of the well-known compactness lemma of Benci and Cerami to this variable exponent setting, and use it to prove that the Palais-Smale condition holds at all level below a certain threshold. We deduce the existence of a ground state when the variable exponent slowly approaches the limit at infinity from below.
The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_λ)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.
We deal with the existence of solutions having L2 regularity for a class of non autonomous evolution equations. Associated with the equation, a general non local condition is studied. The technique we used combines a finite dimensional reduction together with the Leray-Schauder continuation principle. This approach permits to consider a wide class of nonlinear terms by allowing demicontinuity assumptions on the nonlinearity.
A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational…
We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form \[Lu(x)=\sum a_{ij}\partial_{ij}u+{\rm PV}\int_{\R^n}(u(x)-u(x+y))K(y)dy.\] These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels $K$ satisfying that $K(y)|y|^{n+σ}$ is nondecreasing along rays from the origin, for some $σ\in(0,2)$ in case $a_{ij}\equiv0$ and for $σ=2$ in case that $(a_{ij})$ is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for $L$ in star-shaped domains with c…
A non-homogeneous mixed local and nonlocal problem in divergence form is investigated for the validity of the global Calderón-Zygmund estimate for the weak solution to the Dirichlet problem of a nonlinear elliptic equation. We establish an optimal Calderón-Zygmund theory by finding not only a minimal regularity requirement on the mixed local and nonlocal operators but also a lower level of geometric assumption on the boundary of the domain for the global gradient estimate. More precisely, assuming that the nonlinearity of the local operator, whose prototype is the classical $(-Δ_p)^1$-Laplace …
Infinitely many solutions for $p$-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method
In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $\mathbb{L}_γ^s u = \mathcal{F}(u)$, posed in a $C^2$ bounded domain $Ω\subset \mathbb{R}^N$, with appropriate homogeneous boundary or exterior Dirichlet conditions. The operator $\mathbb{L}_γ^s$ belongs to a general class of nonlocal operators including typical fractional Laplacians such as restricted fractional Laplacian, censored fractional Laplacian and spectral fractional Laplacian. The nonlinear term $\mathcal{F}(u)$ covers three different amalgamation of …
In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ Δ})^{\fracα{2}}u- γ\frac{u}{|x|^α}= λf(x) |u|^{q - 2} u + g(x) {\frac{|u|^{p-2}u}{|x|^s}} \ \text{ in } {Ω,} \quad \text{ with Dirichlet boundary condition } u = 0 \ \text{ in } \mathbb{R}^n \setminus Ω,$$ where $Ω\subset \mathbb{R}^n$ is a smooth bounded domain in $\mathbb{R}^n$ containing $0$ in its interior, and $f,g \in C(\overlineΩ)$ with $f^+,g^+ \not\equiv 0$ which may change sign in $\overlineΩ.$ We use the var…
We study an initial value problem with fractional Laplacian and a singular drift term, and obtain local and global existence theorems similar to the results in Jourdain et al.(2005).
We consider the fractional elliptic inequality with variable-exponent nonlinearity $$ (-Δ)^{\fracα{2}} u+λ\, Δu \geq |u|^{p(x)}, \quad x\in\mathbb{R}^N, $$ where $N\geq 1$, $α\in (0,2)$, $λ\in\mathbb{R}$ is a constant, $p: \mathbb{R}^N\to (1,\infty)$ is a measurable function, and $(-Δ)^{\fracα{2}}$ is the fractional Laplacian operator of order $\fracα{2}$. A Liouville-type theorem is established for the considered problem. Namely, we obtain sufficient conditions under which the only weak solution is the trivial one. Next, we extend our study to systems of fractional elliptic inequalities with …
We prove an existence result for nonlinear diffusion equations in the presence of a nonlocal density-dependent drift which is not necessarily potential. The proof is constructive and based on the Helmholtz decomposition of the drift and a splitting scheme. The splitting scheme combines transport steps by the divergence-free part of the drift and semi-implicit minimization steps à la Jordan-Kinderlherer-Otto to deal with the potential part.
We prove the comparison principle for viscosity sub and super solutions of degenerate nonlocal operators with general nonlocal gradient nonlinearities. The proofs apply to purely Hamilton-Jacobi equations of order $0<s<1$.
We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional $p$-Laplacian operator and singular nonlinearities.
We study the existence and nonexistence of normalized solutions $(u_a, λ_a)\in H^{1}(\mathbb{R}^N)\times \mathbb{R}$ to the nonlinear Schrödinger equation with mixed nonlocal nonlinearities. This study can be viewed as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions to the nonlocal Schrödiger equation with a fixed $L^2$-norm $\|u\|_2=a>0$. The leading term is $L^2$-supercritical, that is, $p\in (\frac{N+α+2}{N},\frac{N+α}{N-2}]$, where the Hardy-Littlewood-Sobolev critical exponent $p=\frac{N+α}{N-2}$ appears. We first prove that there exist two normalized …
We investigate the existence of self-similar solutions for a coagulation equation with nonlocal drift. In addition to explicitly given exponentially decaying solutions we establish the existence of self-similar profiles with algebraic decay.
We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth.
In this paper, we are concerned with static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution $u$ in the critical case and non-existence of nontrivial nonnegative solutions in the subcritical cases (see Theorem \ref{Thm0} and \ref{Thm1}). The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians in \cite{CLZ}. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., \emph{Narrow region princip…
We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a "Newtonian" nonlocal term.
We investigate the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial Hölder regularity, we prove that the vanishing viscosity solution is differentiable with some Hölder continuous derivatives for any positive time.
We study the free boundary of the porous medium equation with nonlocal drifts in dimension one. Under the assumption that the initial data has super-quadratic growth at the free boundary, we show that the solution is smooth in space and $C^{2,1}_{\loc}$ in time, and then the free boundary is $C^{2,1}_{\loc}$. Moreover if the drift is local, both the solution and the free boundary are smooth.
We study the fractional laplacian problem (-Δ)^s u &=& u^p -εu^q \quad\text{in }\quad Ω, u &\in& H^s(Ω)\cap L^{q+1}(Ω),u &>&0 \quad\text{in }\quad Ω, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminusΩ, where $s\in(0,1)$, $q>p\geq \frac{N+2s}{N-2s}$ and $ε>0$ is a parameter. Here $Ω\subseteq\mathbb{R}^N$ is a bounded star-shaped domain with smooth boundary and $N> 2 s$. We establish existence of a variational positive solution $u_ε$ and characterize the asymptotic behaviour of $u_ε$ as $ε\to 0$. When $p=\frac{N+2s}{N-2s}$, we describe how the solution $u_ε$ concentrates and blows up at a interior…
We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples including reaction-diffusion systems with nonlocal quadratic nonlinearities and the nonlinear Schrodinger equation with a…
We investigate the NLS equation with competing Hartree-type and power-type nonlinearities \begin{equation*} \begin{array}{ll} i\partial _{t}ψ+Δψ+γ(I_{α}\ast |ψ|^{p})|ψ|^{p-2}ψ+μ|ψ|^{q-2}ψ=0, & \text{ }\forall (t,x)\in \mathbb{R\times R}^{N},% \end{array}% \end{equation*}% where $γμ<0$. We establish conditions for the local well-posedness in the energy space. Under the double $L^{2}$-supercritical case, we prove the existence and multiplicity of standing waves with prescribed mass by developing a constraint method when $γ<0,μ>0$ and $γ>0,μ<0, $ respectively. Moreover, we prove weak orbital stab…
We provide a suitable variational approach for a class of nonlocal problems involving the fractional laplacian and singular nonlinearities for which the standard techniques fail. As a corollary we deduce a characterization of the solutions.
We consider the nonlinear Schroedinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a "small divisor problem" which we do by using a renormalisation group approach.
We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the sup…
This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case. where the drift is a gradient (in the physical space), we obtain existence by a semi-implicit Jordan-Kinderlehrer-Otto scheme. Then, in the nonpotential case, we derive existence from a regularization procedure and parabolic energy estimates. We also address the uniqueness issue by a displacement convexity argument.
We prove an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications are given to subcritical and critical $p$-Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional $p$-Laplacian problems.
We consider a one-dimensional nonlocal nonlinear equation of the form: $\partial_t u = (Λ^{-α} u)\partial_x u - νΛ^βu$ where $Λ=(-\partial_{xx})^{\frac 12}$ is the fractional Laplacian and $ν\ge 0$ is the viscosity coefficient. We consider primarily the regime $0<α<1$ and $0\le β\le 2$ for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D $α$-patch models. In the critical and subcritical range $1-α\le β\le 2$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $0 \le β<1-α$, we …
In \cite{CordobaCordobaFontelos05}, Córdoba, Córdoba, and Fontelos proved that for some initial data, the following nonlocal-drift variant of the 1D Burgers equation does not have global classical solutions \[ \partial_t θ+u \; \partial_x θ= 0, \qquad u = H θ, \] where $H$ is the Hilbert transform. We provide four essentially different proofs of this fact. Moreover, we study possible Hölder regularization effects of this equation and its consequences to the equation with diffusion \[ \partial_t θ+ u \; \partial_x θ+ Λ^γθ= 0, \qquad u = H θ, \] where $Λ= (-Δ)^{1/2}$, and $1/2 \leq γ<1$. Our res…
