In this work, we prove some trace theorems for function spaces with a nonlocal character that contain the classical $W^{s,p}$ space as a subspace. The result we obtain generalizes well known trace theorems for $W^{s,p}(Ω)$ functions which has a well-defined $W^{1-s/p, p}$ trace on the boundary of a domain with sufficient regularity. The new generalized spaces are associated with norms that are characterized by nonlocal interaction kernels defined heterogeneously with a special localization feature on the boundary. Intuitively speaking, the class contains functions that are as rough as an $L^p$ function inside the domain of definition but as smooth as a $W^{s,p}$ function near the boundary. Our result is that the $W^{1-s/p, p}$ norm of the trace on the boundary such functions is controlled by the nonlocal norms that are weaker than the classical $W^{s, p}$ norm. These results are improvement and refinement of the classical results since the boundary trace can be attained without imposing regularity of the function in the interior of the domain. They also extend earlier results in the case of $p=2$. In the meantime, we prove Hardy-type inequalities for functions in the new generalized spaces that vanish on the boundary, showing them having the same decay rate to the boundary as functions in the smaller space $W^{s,p}(Ω)$. A Poincaré-type inequality is also derived. An application of the new theory leads to the well-posedness of a nonlinear variational problem that allows possible singular behavior in the interior with imposed smoother data on the boundary.
By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.
We prove Hardy's inequality for the fractional powers of the generalized sublaplacian and the fractional powers of the Grushin operator. We also find an integral representation and a ground state representation for the fractional powers of generalized sublaplacian.
In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.
Mixed local-nonlocal quasilinear problems with mixed interpolated Hardy potential
Ghent University（根特大学）；National Institute of Technology Calicut（卡利卡特国立技术学院）；Indian Institute of Technology (BHU)（印度理工学院（贝拿勒斯））；Queen Marry University of London（伦敦玛丽女王大学）
We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.
Our main aim is to study Green function for the fractional Hardy operator $P:=(-Δ)^s -\fracθ{|x|^{2s}}$ in $\mathbb{R}^N$, where $0<θ<Λ_{N,s}$ and $Λ_{N,s}$ is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds.
Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.
A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren-Poon monotonicity formula combined with a blow-up analysis.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$.
The present work is concerned with existence of positive solutions for a class of fractional equation involving a Kirchhoff term and singular potential.
We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]
We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λf(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} Ω, $$ where $Ω$ is a smooth bounded domain of $\mathbb{R}^N$, $λ>0$, $N\geq3$, $0<μ<N$ and $2_μ^{\ast}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities $f(u)$, we are able to prove some existence and multiplicity results for the equation by variational methods.
We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.
In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left\{% \begin{array}{ll} (-Δ)^{s} u-\ds\frac{μu}{|x|^{2s}}=|u|^{2^*_s-2}u+au, & \hbox{$\text{in}~ Ω$},\vspace{0.1cm} u=0,\,\, &\hbox{$\text{on}~\partial Ω$}, \\ \end{array}% \right. \end{equation*} provided $N>6s$, $μ\geq0$, $0< s<1$, $2^*_s=\frac{2N}{N-2s}$, $a>0$ is a constant and $Ω$ is an open bounded domain in $\R^N$ which contains the origin.
The general stability problem of truncations for a family of functions concentrating mass at the origin is described and a concrete example in the framework of entire optimizers for the fractional Hardy-Sobolev inequality is given. In this short note we point out some quantitative stability estimates, useful in dealing with critical $p-q$ fractional equations.
We prove the existence, qualitative properties and asymptotic behavior of positive solutions to the doubly critical problem $$ (-Δ)^s u=\vartheta\frac{u}{|x|^{2s}}+u^{2_s^*-1}, \quad u\in \dot{H}^s(\mathbb{R}^N).$$ The technique that we use to prove the existence is based on variational arguments. The qualitative properties are obtained by using of the moving plane method, in a nonlocal setting, on the whole $\mathbb{R}^N$ and by some comparison results. Moreover, in order to find the asymptotic behavior of solutions, we use a representation result that allows to transform the original problem…
Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of $\R^d$ are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
We obtain existence, multiplicity, and bifurcation results for the Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator, involving critical Hardy-Sobolev exponents. Our results are mainly extend results in the literature for $α=0$. In the absence of an explicit formula for a minimizer in the fractional Hardy-Sobolev inequality $α\not= 0$, we get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in \cite{MM}.
The paper deals with existence and multiplicity of positive solutions to nonlocal equations with critical Hrardy-Sobolev nonlinearities and external terms. We establish the profile decomposition of the Palais-Smale sequences associated with the functional and existence of at least two positive solutions to the equation.
In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the Fractional Cauchy problem with the Hardy potential, namely, \begin{equation*} u_t+(-Δ)^s u=λ\dfrac{u}{|x|^{2s}}+u^{p}\inn\ren,\\ u(x,0)=u_{0}(x)\inn\ren, \end{equation*} where $N> 2s$, $0<s<1$, $(-Δ)^s$ is the fractional laplacian of order $2s$, $ł>0$, $u_0\ge 0$, and $1<p<p_{+}(s,λ)$, where $p_{+}(λ, s)$ is the critical existence power found in \cite{BMP} and \cite{AMPP}.
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…
In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-Δ)^{\frac α2} u=λ|u|^{r-2}u+μ\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in }\quad Ω, \\ \\ u=0 &{\rm on }\quad \partialΩ, \end{array} \right. \end{equation} where $Ω$ is a smooth bounded domain in $\mathbb R^N$ with $0\in Ω$, $λ,μ>0,0<s\leqα$, $(-Δ)^{\frac α2}$ is the fractional Laplacian operator with $0<α<2$. We establish existence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
This article concerns about the existence and multiplicity of weak solutions for the following nonlinear doubly nonlocal problem with critical nonlinearity in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{ \begin{split} (-Δ)^su &= λ|u|^{q-2}u + \left(\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y\right) |u|^{2^*_μ-2}u\; \text{in}\; Ω (-Δ)^sv &= δ|v|^{q-2}v + \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}~\mathrm{d}y \right) |v|^{2^*_μ-2}v \; \text{in}\; Ω u &=v=0\; \text{in}\; \mb R^n\setminusΩ, \end{split} \right. \end{equation*} where $Ω$ is a smooth bounded dom…
In this paper, we deal with a fractional elliptic equation with critical Sobolev nonlinearity and Hardy term $$ (-Δ)^α u-μ\frac{u}{|x|^{2α}}+a(x) u=|u|^{2^*-2}u+k(x)|u|^{q-2}u$$ $$ u\,\in\,H^α({\mathbb R}^N),$$ where $2<q< 2^*$, $0<α<1$, $N>4α$, $2^*=2N/(N-2α)$ is the critical Sobolev exponent, $a(x),k(x)\in C({\mathbb R}^N)$. Through a compactness analysis of the functional associated to $(*)$, we obtain the existence of positive solutions for $(*)$ under certain assumptions on $a(x),k(x)$.
In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $λ>0$ we analyze the attainability of the optimal constant $$ μ_{α, λ}(Ω):=\inf\left\{ [u]^2_{s,Ω}+λ\int_Ω|u|^2 \, dx \colon u\in H^s(Ω), \, \int_Ω \frac{|u(x)|^{2_{s,α}}}{|x|^α} \, dx=1 \right\}, $$ where $0<s<1, n>4s, 0<α<2s$, $2_{s,α}=\frac{2(n-α)}{n-2s}$, and $Ω\subset \mathbb{R}^n$ be a bounded domain such that $0\in Ω$.
In this paper, we consider the fractional elliptic equation \begin{align*} \left\{\begin{aligned} &(-Δ)^s u-μ\frac{u}{|x|^{2s}} = \frac{|u|^{2_s^\ast (α)-2}u}{|x|^α} + f(x,u), && \mbox{in} \ Ω,\\ &u=0, && \mbox{in} \ \mathbb{R}^{n}\backslash \ Ω, \end{aligned}\right. \end{align*} where $Ω\subset R^n$ is a smooth bounded domain, $0\inΩ$, $0<s<1$, $0<α<2s<n$, $2_{s}^{\ast}(α)=\frac{2(n-α)}{n-2s}$. Under some assumptions on $μ$ and $f$, we obtain the existence of nonnegative solutions.
The purpose of this paper is to study and classify singular solutions of the Poisson problem $$ %\begin{equation}\label{eq 0.1} \left \{ \begin{aligned} {\mathcal L}^s_μu = f \quad\ {\rm in}\ \, Ω\setminus \{0\},\\ u =0 \quad\ {\rm in}\ \, {\mathbb R}^N \setminus Ω %\\ %\liminf_{x \to 0}\:|u(x)| /Φ_μ(x) = k. \end{aligned} \right. $$ for the fractional Hardy operator ${\mathcal L}_μ^s u= (-Δ)^s u +\fracμ{|x|^{2s}}u$ in a bounded domain $Ω\subset {\mathbb R}^N$ ($N \ge 2$) containing the origin. Here $(-Δ)^s$, $s\in(0,1)$, is the fractional Laplacian of order $2s$, and $μ\ge μ_0$, where $μ_0 = -…
