在维度$N\geq 5$且$0<s<4$及$\gamma\in\mathbb{R}$下，我们研究双重临界问题$$\Delta^2 u-\frac{\gamma}{|x|^4}u= |u|^{2_{0}^{\star}-2}u+\frac{|u|^{2_{s}^{\star}-2}u}{|x|^s}\hbox{ in }\mathbb{R}_+^N,\\; u=\Delta u=0\hbox{ on }\partial \mathbb{R}_+^N,$$的非平凡弱解存在性，其中$2_{s}^{\star}:=\frac{2(N-s)}{N-4}$是临界Hardy-Sobolev指数。对于$N\geq 8$且$0<\gamma<\frac{(N^2-4)^2}{16}$，我们利用Ambrosetti-Rabinowitz的Mountain-Pass定理证明了非平凡解的存在性。所用方法基于本文证明的某些Hardy-Sobolev嵌入极值的存在性。
英文摘要：
In dimension $N\geq 5$, and for $0<s<4$ with $γ\in\mathbb{R}$, we study the existence of nontrivial weak solutions for the doubly critical problem
$$Δ^2 u-\fracγ{|x|^4}u= |u|^{2_{0}^{\star}-2}u+\frac{|u|^{2_{s}^{\star}-2}u}{|x|^s}\hbox{ in }\mathbb{R}_+^N,\; u=Δu=0\hbox{ on }\partial \mathbb{R}_+^N,$$ where $2_{s}^{\star}:=\frac{2(N-s)}{N-4}$ is the critical Hardy-Sobolev exponent. For $N\geq 8$ and $0<γ<\frac{(N^2-4)^2}{16}$, we show the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti-Rabinowitz. The method used is based on the existence of extremals for certain Hardy-Sobolev embeddings that we prove in this paper.
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…
We study weighted inequalities of Hardy and Hardy-Poincaré type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile.
By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-type inequalities: \begin{equation*} \int_{\mathbb{R}^N}|x|^α|Δu|^2 dx \geq S^{rad}(N,α)\left(\int_{\mathbb{R}^N}|x|^{-α}|u|^{p^*_α} dx\right)^{\frac{2}{p^*_α}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<α<2$, $p^*_α=\frac{2(N-α)}{N-4+α}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{λ,α}$ to the weighted fourth-order Hardy (for $α>0$) or…
We examine the following fourth order Hénon equation \label{pipe} Δ^2 u = |x|^αu^p \qquad \text{in}\ \IR^N, where $ 0 < α$. Define the Hardy-Sobolev exponent $ p_4(α):= \frac{N+4 + 2 α}{N-4}$. We show that in dimension N=5 there are no positive bounded classical solutions of (\ref{pipe}) provided $ 1 < p < p_4(α)$.
This paper studies a fourth-order, nonlinear, doubly-degenerate parabolic equation derived from the thin film equation in spherical geometry. A regularization method is used to study the equation and several useful estimates are obtained. The main result of this paper is to prove the existence of a weak solution of the equation in a weighted Sobolev space.
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…
Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to weighted doubly $D^{1,p}$-critical quasi-linear nonlocal elliptic equations with Hardy potential
In this paper, we consider the relation between $p > 1$ and critical dimension of the extremal solution of the semilinear equation $$\{\begin{array}{lllllll} βΔ^{2}u-τΔu=\fracλ{(1-u)^{p}} & in\ \ B, 0<u\leq 1 & in\ \ B, u=Δu=0 & on\ \ \partial B, \end{array} . $$ where $B$ is the unit ball in $R^{n}$, $λ>0$ is a parameter, $τ>0, β>0,p>1$ are fixed constants. By Hardy-Rellich inequality, we find that when $p$ is large enough, the critical dimension is 13.}
We give necessary and sufficient conditions on a pair of positive radial functions $V$ and $W$ on a ball $B$ of radius $R$ in $R^{n}$, $n \geq 1$, so that the following inequalities hold \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x) u^{2}dx+b\int_{\partial B}u^2 ds$ for all u $\in H^1(B)$,} \end{equation*} and \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|Δu |^{2}dx \geq \int_{B} W(x)|\nabla u|^{2}dx+b\int_{\partial B}|\nabla u|^2 ds$ for all u $\in H^2(B)$.} \end{equation*} Then we present various classes of optimal weighted Hardy-Rellich inequaliti…
We study the regularity of the extremal solution of the semilinear biharmonic equation $βΔ^2 u-τΔu=\fracλ{(1-u)^2}$ on a ball $B \subset \R^N$, under Navier boundary conditions $u=Δu=0$ on $\partial B$, where $λ>0$ is a parameter, while $τ>0$, $β>0$ are fixed constants. It is known that there exists a $λ^{*}$ such that for $λ>λ^{*}$ there is no solution while for $λ<λ^{*}$ there is a branch of minimal solutions. Our main result asserts that the extremal solution $u^{*}$ is regular ($\sup_{B}u^{*}<1$) for $N\leq 8$ and $β, τ>0$ and it is singular ($\sup_{B}u^{*}=1$) for $N\geq 9$, $β>0$, and $τ…
In this article, we study the following nonlinear doubly nonlocal problem involving the fractional Laplacian in the sense of Hardy-Littlewood-Sobolev inequality \begin{equation*} \left\{\begin{aligned} (-Δ)^s u & = au+bv+\frac{2p}{p+q}\int_Ω\frac{|v(y)|^q}{|x-y|^μ}dy|u|^{p-2}u+2ξ_1\int_Ω\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy|u|^{2^*_μ-2}u,&& \text{in } Ω;\\ (-Δ)^s v & = bu+cv+\frac{2q}{p+q}\int_Ω\frac{|u(y)|^p}{|x-y|^μ}dy|v|^{q-2}v+2ξ_2\int_Ω\frac{|v(y)|^{2^*_μ}}{|x-y|^μ}dy|v|^{2^*_μ-2}v,&& \text{in } Ω;\\ u &=v=0,\text{ in } \R^N\setminusΩ, \end{aligned}\right. \end{equation*} where $Ω$ is a smooth…
In this paper, we are devoted to studying the positive solutions of the following higher order Hardy-Hénon equation $$ (-Δ)^{m}u=|x|^αu^{p} \quad\mbox{in}~ B_{1}\setminus\{0\}\subset\mathbb{R}^{n} $$ with an isolated singularity at the origin, where $α>-2m$, $m\geq1$ is an integer and $n>2m$. For $1<p<\frac{n+2m}{n-2m}$, singularity and decay estimates of solutions will be given. For $\frac{n+α}{n-2m}<p<\frac{n+2m}{n-2m}$ with $-2m<α<2m$, we show the super polyharmonic properties of solutions near the singularity, which are essential tools in the study of polyharmonic equation. Using these pro…
Some qualitative and quantitative properties of weak solutions to mixed anisotropic and nonlocal quasilinear elliptic and doubly nonlinear parabolic equations
In this paper, we are concerned with the following Schrödinger-Poisson system with critical nonlinearity and critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality \begin{equation}\begin{cases} -Δu+u+λϕ|u|^3u =|u|^4u+ |u|^{q-2}u,\ \ &\ x \in \mathbb{R}^{3},\\[2mm] -Δϕ=|u|^5, \ \ &\ x \in \mathbb{R}^{3}, \end{cases} \end{equation} where $λ\in \mathbb{R}$ is a parameter and $q\in(2,6)$. If $λ\ge (\frac{q+2}{8})^2$ and $q\in(2,6)$, the above system has no nontrivial solution. If $λ\in (λ^*,0)$ for some $λ^*<0$, we obtain a least energy radial sign-changing solution $u_λ$ to the abo…
We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions wit…
We study existence and long-time behaviour of strong solutions for the thin film equation using a priori estimates in a weighted Sobolev space. This equation can be classified as a doubly degenerate fourth-order parabolic and it models coating flow on the outer surface of a sphere. It is shown that the strong solution asymptotically decays to the flat profile.
We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\leq s<α<2$, $n>α$, ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ and $γ\in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\R^n$, \beg…
We prove second and fourth order improved Poincaré type inequalities on the hyperbolic space involving Hardy-type remainder terms. Since theirs l.h.s. only involve the radial part of the gradient or of the laplacian, they can be seen as stronger versions of the classical Poincaré inequality. We show that such inequalities hold true on model manifolds as well, under suitable curvature assumptions and sharpness of some constants is also discussed.
We prove the existence of at least two doubly periodic vortex solutions for a self-dual CP(1) Maxwell-Chern-Simons model. To this end we analyze a system of two elliptic equations with exponential nonlinearities. Such a system is shown to be equivalent to a fourth-order elliptic equation admitting a variational structure.
We study the gradient-flow structure of a non-Newtonian thin film equation with power-law rheology. The equation is quasilinear, of fourth order and doubly-degenerate parabolic. By adding a singular potential to the natural Dirichlet energy, we introduce a modified version of the thin-film equation. Then, we set up a minimising-movement scheme that converges to global positive weak solutions to the modified problem. These solutions satisfy an energy-dissipation equality and follow a gradient flow. In the limit of a vanishing singularity of the potential, we obtain global non-negative weak solu…
