In this paper we study the problem, posed by Troyanov, of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function $K$ changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, by a min-max approach and a new compactness argument we give sufficient conditions on $K$, concerning mainly the regularity of its nodal line and the topology of its positive nodal region, to be the Gaussian curvature of a conformal metric with assigned conical singularities.
Besides, we find a class of functions on $\mathbb{S}^2$ which do not verify our conditions and which can not be realized as the Gaussian curvature of any conformal metric with one conical singularity. This shows that our result is somehow sharp.
We study the asymptotic behaviors of solutions of the Loewner-Nirenberg problem in singular domains and prove that the solutions are well approximated by the corresponding solutions in tangent cones at singular points on the boundary. The conformal structure of the underlying equation plays an essential role in the derivation of the optimal estimates.
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in domains with conic singularities and establish asymptotic expansions with respect to two normal directions simultaneously. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Key step is the study of the eigenvalues growth and eigenfunctions estimates.
In this paper, using Mountain Pass Lemma and Linking Argument, we prove the existence of nontrivial weak solutions for the Dirichlet problem for the superlinear equation of Caffarelli-Kohn-Nirenberg type in the case where the parameter $λ\in (0, λ_2)$, $λ_2$ is the second positive eigenvalue of the quasilinear elliptic equation of Caffarelli-Kohn-Nirenberg type.
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in finite cones and establish optimal asymptotic expansions in terms of the corresponding solutions in infinite cones. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Due to the singularity of the spherical domains, extra cares are needed for the study of the global regularity of the eigenfunctions and solutions of the associated singular Dirichlet problem.
In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form…
We consider the Brezis--Nirenberg problem for the Laplacian with a singular drift for a (geodesic) ball in both $\mathbb{R}^{n}$ and $\mathbb{S}^n$, $3 \le n \le 5$. The singular drift we consider derives from a potential which is symmetric around the center of the (geodesic) ball. Here the potential is given by a parameter ($δ$ say) times the logarithm of the distance to the center of the ball. In both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions. The parameter space i…
This paper is a continuation of our earlier work "[T. Jin, Y.Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, to appear in J. Eur. Math. Soc.]", where compactness results were given on a fractional Nirenberg problem. We prove two existence results stated there. We also obtain a fractional Aubin inequality.
In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-Δ)^{s} u-μ\frac{u}{|x|^{2s}}-λu=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} Ω,\ \ \, u=0\,&\text{in} \R^N\backslashΩ \end{cases} \eqno {(*)} $$ where $s\in (0,1)$, $2^*_{s}=2N/(N-2s)$ is the critical Sobolev exponent, $f(x,u)$ is a lower order perturbation of critical Sobolev nonlinearity. We obtain the existence of the solution for (*) through variational methods. In particular we derive a Brézis-Nirenberg type result when $f(x,u)=0$.
We develop a new method for estimating the region of the spectral parameter of a generalized Brezis--Nirenberg problem for which there are no, non trivial, smooth solutions. This new method combines the standard Rellich--Pohozaev argument with a Hardy type inequality for bounded domains. The estimates we get are better than the usual estimates for low dimensions.
In this paper we establish existence results for the Fractional Nirenberg Equation via the flatness hypothesis. We have been able to include the Morse functions in our study. This extends the previous results obtained bY Yan Yan Li and Coauthors. We also discuss the loss of compactness. Our method is self-contained and hinges on the breakthrough results of Bahri andBahri and Coron.
The present paper studies the non-local fractional analogue of the famous paper of Brezis and Nirenberg in [4]. Namely, we focus on the following model, $$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-Δ\right)^s u-λu &= α|u|^{p-2}u + β|u|^{2^*-2}u \quad\mbox{in}\quad Ω,\\ u&=0\quad\mbox{in}\quad\mathbb{R}^N\setminusΩ, \end{cases} \end{align*}$$ where $(-Δ)^s$ is the fractional Laplace operator, $s \in (0,1)$, with $N \geq 3s$, $2<p<2^*$, $β>0, λ, α\in \mathbb{R}$ and establish the existence of nontrivial solutions and sign-changing solutions for the problem $(\mathcal{P})$.
We prove a Caffarelli--Kohn--Nirenberg inequality in the hyperbolic space. For a semilinear elliptic equation involving the associated weighted Laplace--Beltrami operator, we establish variationally the existence of positive radial solutions in the subcritical regime. We also show a non-existence result in star-shaped domains when the exponent is supercritical.
By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.
In this paper, we study the existence of multiple solutions to a Caffarelli-Kohn-Nirenberg type equation with asymptotically linear term at infinity. In this case, the well-known Ambrosetti-Rabinowtz type condition doesn't hold, hence it is difficult to verify the classical (PS)$_c$ condition. To overcome this difficulty, we use an equivalent version of Cerami's condition, which allows the more general existence result.
We consider the fractional Nirenberg problem on the standard sphere $\mathbb{S}^n$ with $n\geq 4$. Using the theory of critical points at infinity, we establish an Euler-Hopf type formula and obtain some existence results for curvature satisfying assumptions of Bahri-Coron type.
In the present paper, we study the effect of an elliptic operator in divergence form L defined in a bounded open set on the existence and nonexistence of positive solutions of the associated nonlocal Brezis-Nirenberg problem involving powers of L.
In this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness properties of solutions.
This paper is devoted to the existence of positive solutions for a problem related to a fourth-order differential equation involving a nonlinear term depending on a second order differential operator, $$(-Δ)^2 u=λu+ (-Δ)|u|^{p-1}u,$$ in a bounded domain $Ω\subset\mathbb{R}^N$, $N\geq 7$, and assuming homogeneous Navier boundary conditions. In particular, we study a second order equation involving a nonlocal term of the form, $$-Δu=λ(-Δ)^{-1} u+|u|^{p-1}u,$$ under Dirichlet boundary conditions and we prove the existence of positive solutions depending on the positive real parameter $λ>0$, up to…
We prove some existence and nonexistence results for a class of critical $(p,q)$-Laplacian problems in a bounded domain. Our results extend and complement those in the literature for model cases.
The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.
In this paper we extend the well-known concentration -- compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional $p-$laplacian in the whole ${\mathbb R}^n$.
Using a Rellich-Pohozaev argument and Hardy's inequality, we derive an improved bound on the nonlinear eigenvalue for the non existence of radial solutions of a Brezis-Nirenberg problem, with Dirichlet boundary conditions, on a geodesic ball of Hn, for 2<n<4.
We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-Δu =\left(\int_Ω\frac{|u|^{2_μ^{\ast}}}{|x-y|^μ}dy\right)|u|^{2_μ^{\ast}-2}u+λu\4.14mm\mbox{in}\1.14mm Ω, $$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz boundary, $λ$ is a real parameter, $N\geq3$, $2_μ^{\ast}=(2N-μ)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
Mixed-dispersion Schrödinger equations and Gagliardo-Nirenberg inequalities: equivalence between ground states and optimizers
China University of Geosciences（中国地质大学）；Institute for Advanced Marine Research, China University of Geosciences（中国地质大学高级海洋研究院）；Università degli Studi di Udine（乌迪内大学）；Anhui University of Science and Technology（安徽理工大学）
We consider the problem of finding on a given Euclidean domain $Ω$ of dimension $n \geq 3$ a complete conformally flat metric whose Schouten curvature $A$ satisfies some equation of the form $f(λ(-A)) = 1$. This generalizes a problem considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of such metric when the boundary $\partialΩ$ is a smooth bounded hypersurface (of codimension one). When $\partialΩ$ contains a compact smooth submanifold $Σ$ of higher codimension with $\partialΩ\setminusΣ$ being compact, we also give a `sharp' condition for the di…
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}.
This paper is devoted to study a class of integral type Brezis-Nirenbreg problem on the Heisenberg group. It is a class of new nonlinear integral equations on the bounded domains of Heisenberg group and related to the CR Yamabe problems on the CR manifold. Based on the sharp Hardy-Littlewood-Sobolev inequalities, the nonexistence and existence results are obtained by Pohozaev type identity, variational method and blow-up analysis, respectively.
The main purpose of this paper is to establish the existence, nonexistence and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg problems associated with the GJMS operators $P_k$ on bounded domains in the hyperbolic space $\mathbb{H}^n$ and as well as on the entire hyperbolic space $\mathbb{H}^n$. Among other techniques, one of our main novelties is to use crucially the Helgason-Fourier analysis on hyperbolic spaces and the higher order Hardy-Sobolev-Maz'ya inequalities and careful study of delicate properties of Green's functions of $P_k-λ$ on hyperbolic spaces which are o…
Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\it (Compositio Mathematica,1984):} $$\Big(\int_Ω\frac{|u|^r}{|x|^s}dx\Big)^{\frac{1}{r}}\leq C(p,q,r,μ,σ,s)\Big(\int_Ω\frac{|\nabla u|^p}{|x|^μ}dx\Big)^{\frac{a}{p}}\Big(\int_Ω\frac{|u|^q}{|x|^σ}dx\Big)^{\frac{1-a}{q}},$$ where $Ω\subset \R^N (N\geq 2)$ is an open set; $p, q, r, μ, σ, s, a$ are some parameters satisfying some balanced conditions. When $Ω$ is a cone in $\R^N$ (for example, $Ω=\R^N)$, we prove the sharp constant $C(p,q,r,μ,σ,s)$ can be achieved for a very large parameter space. Besides, we find some suff…
In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $n=3,$ $σ=1/2,$ when the prescribing $σ$-curvature function satisfies the $(n-2σ)$-flatness condition near its critical points. The compactness results are new and optimal. In addition, we obtain a degree-counting formula of all solutions. From our results, we can know where blow up occur. Moreover, for any finite distinct points, the sequence of solutions that blow up precisely at these points can be constructed. We extend the results of Li in \cite[CPAM…
We establish theorems on the existence and compactness of solutions to the $σ_2$-Nirenberg problem on the standard sphere $\mathbb S^2$. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order der…
We study the existence/nonexistence of positive solution of $$ {Δ^2u-μ\frac{u}{|x|^4}=\frac{|u|^{q_β-2}u}{|x|^β}\quad\textrm{in $Ω$,}} $$ when $Ω$ is a bounded domain and $N\geq 5$, $q_β=\frac{2(N-β)}{N-4}$, $0\leq β<4$ and $0\leqμ<\big(\frac{N(N-4)}{4}\big)^2$. We prove the nonexistence result when $Ω$ is an open subset of $\mathbf R^N$ which is star shaped with respect to the origin. We also study the existence of positive solution in $Ω$ when $Ω$ is a bounded domain with non trivial topology and $β=0$, $μ\in(0,μ_0)$, for certain $μ_0<\big(\frac{N(N-4)}{4}\big)^2$ and $N\geq 8$. Different be…
In this paper, the fractional order curvature equation $(-Δ)^γu = (1 + \varepsilon K(x))u^{\frac{N + 2γ}{N - 2γ}}$ in $\mathbb{R}^N$ is considered. Assuming $K(x)$ has two critical points satisfying certain local conditions, we prove the existence of two-peak solutions.
Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case
We consider the problem of prescribing the $σ_k$-curvature on the standard sphere $\mathbb{S}^n$ with $n \geq 3$. We prove existence and compactness theorems when $k \geq n/2$. This extends an earlier result of Chang, Han and Yang for $n = 4$ and $k = 2$.
In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., $$ \left\{\begin{array}{rcl} (-Δ)^su & = & λu+u^{2_s^*-1},\quad u>0\quad\mbox{in}\quad Ω,\\ u & = & 0\quad\mbox{on}\quad Σ_{\mathcal{D}},\\ \displaystyle\frac{\partial u}{\partial ν} & = & 0\quad\mbox{on}\quad Σ_{\mathcal{N}}, \end{array}\right. $$ where $Ω\subset\mathbb{R}^N$ is a regular bounded domain, $\frac{1}{2}<s<1$, $2_s^*$ is the critical fractional Sobolev exponent, $0\leλ\in \math…
We are concerned with existence results for a critical problem of Brezis-Nirenberg Type involving an integro-differential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It hinges on appropriate choices of parameters in the mountain-pass theorem
In this paper we study the following class of nonlocal {problems} involving Caffarelli-Kohn-Nirenberg type critical growth \begin{align*} L(u)&-λh(x)|x|^{-2(1+a)}u=μf(x)|u|^{q-2}u+|x|^{-pb}|u|^{p-2}u\;\; \text{in } \mathbb R^N, \end{align*} where $h(x)\geq 0$, $f(x)$ is a continuous function which may change sign, $λ, μ$ are positive real parameters and $1<q<2$, $4< p=2N/[N+2(b-a)-2]$, $0\leq a<b<a+1<N/2$, $N\geq 3$. Here $$ L(u)=-M\left(\int_{\mathbb R^N} |x|^{-2a}|\nabla u|^2dx\right)\mathrm {div}(|x|^{-2a}\nabla u) $$ and the function $M:\mathbb R^+\cup \{0\} \to\mathbb R^+$ is exactly as i…
We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures $K$ contained in naturally defined stable regions. We prove that in such stable regions, the map $u \rightarrow K_{g}$, $g = e^{2u}g_{+1}$ is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also p…
In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb's Translation Lemma and a Riesz energy version of the Brézis--Lieb lemma.
