Critical Elliptic Systems in Potential Form

Critical Chirality in Elliptic Systems

We establish the regularity in 2 dimensions of $L^2$ solutions to critical elliptic systems in divergence form involving involution operators of finite $W^{1,2}$-energy.

非局部临界哈密顿椭圆系统中无穷多同步解

Infinitely many synchronized solutions for a nonlocal critical Hamiltonian elliptic system

本文研究了一类具有Hartree型非局部相互作用的临界哈密顿椭圆系统的无穷多同步解的存在性问题。

关于带有Hardy-Littlewood-Sobolev临界指数和Sobolev-Hardy临界指数的非线性椭圆问题

On nonlinear elliptic problems with Hardy-Littlewood-Sobolev critical exponent and Sobolev-Hardy critical exponent

本文利用变分方法研究一类涉及混合卷积型和Hardy非线性的非线性椭圆问题，探讨亚临界和临界增长下的解的存在性。

Critical functions and elliptic PDE on compact riemannian manifolds

We study in this work the existence of minimizing solutions to the critical-power type equation $\triangle_{\textbf{g}}u+h.u=f .u^{\frac{n+2}{n-2}} $ on a compact riemannian manifold in the limit case normally not solved by variational methods. For this purpose, we use a concept of "critical function" that was originally introduced by E. Hebey and M. Vaugon for the study of second best constant in the Sobolev embeddings. Along the way, we prove an important estimate concerning concentration phenomena's when $f$ is a non-constant function. We give here intuitive details.

On critical elliptic problems with singular Trudinger-Moser nonlinearities

We establish some existence results for a class of critical elliptic problems with singular exponential nonlinearities. We do not assume any global sign conditions on the nonlinearity, which makes our results new even in the nonsingular case.

一类带临界增长和对数扰动的非线性退化椭圆算子的Dirichlet问题

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

该研究聚焦于一类带临界增长与对数扰动的非线性退化椭圆算子的Dirichlet问题，探究其弱解的存在性。

Nodal solutions for fourth order elliptic equations with critical exponent on compact manifolds

Using a variational method we prove the existence of nodal solutions to prescribed scalar Q- curvature type equations on compact Riemannian manifolds with boundary; these equations are fourth-order elliptic equations with critical Sobolev growth .

Critical sets of elliptic equations

Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\Cr(u)\equiv \{x:|\nabla u|(x)=0\}$. The results are new even for harmonic functions on $\dR^n$. Given such a $u$, the standard {\it first order} stratification $\{\cS^k\}$ of $u$ separates points $x$ based on the degrees of symmetry of the leading order polynomial of $u-u(x)$. In this paper we give a quantitative stratification $\{\cS^k_{η,r}\}$ of $u$, which separates points based on the number of {\it almost} symm…

Nonexistence results for a class of nonlinear elliptic equations involving critical Sobolev exponents

In this paper we study the problem of bifurcation from the origin of solutions of elliptic Dirichlet problems involving critical Sobolev exponent, defined on a bounded domain $Ω$ in $\mathbb{R} ^N$: we prove that the first critical case are $N=3, 4$ (not only N=3, as just proved by Brezis and Nirenberg), exhibiting two nonexistence results for a class of elliptic problem in these dimensions.

Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearities

We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $\mathbb{R}^n$.

Three solutions for a fractional elliptic problems with critical and supercritical growth

In this paper, we deal with the existence and multiplicity of solutions for the fractional elliptic problems involving critical and supercritical Sobolev exponent via variational arguments. By means of the truncation combining with the Moser iteration, we prove that the problems has at least three solutions.

各向异性等周问题近临界点的气泡现象

Bubbling of almost critical points of anisotropic isoperimetric problems with degenerating ellipticity

研究各向异性表面能函数的近临界点在范数退化时的极限行为，证明其极限为有限个不相交的Wulff形状的并集。

Upper bound of critical sets of solutions of elliptic equations in the plane

In this note, we investigate the measure of singular sets and critical sets of real-valued solutions of elliptic equations in two dimensions. These singular sets and critical sets are finitely many points in the plane. Adapting the Carleman estimates involving polynomial functions at singularities by Donnelly and Fefferman in \cite{DF90}, we obtain the upper bounds of singular points and critical points.

A general perturbation theorem with applications to nonhomogeneous critical growth elliptic problems

We prove a general perturbation theorem that can be used to obtain pairs of nontrivial solutions of a wide range of local and nonlocal nonhomogeneous elliptic problems. Applications to critical $p$-Laplacian problems, $p$-Laplacian problems with critical Hardy-Sobolev exponents, critical fractional $p$-Laplacian problems, and critical $(p,q)$-Laplacian problems are given. Our results are new even in the semilinear case $p = 2$.

The critical semilinear elliptic equation with isolated boundary singularities

We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.

Boundary Regularity for an Even Order Elliptic System in the Critical Dimension

In this short note, we consider the Dirichlet problem associated to an even order elliptic system with antisymmetric first order potential. Given any continuous boundary data, we show that weak solutions are continuous up to boundary.

Weak compactness of solutions for fourth order elliptic systems with critical growth

We consider a class of fourth order elliptic systems which include the Euler-Lagrange equations of biharmonic mappings in dimension 4 and we prove that weak limit of weak solutions to such systems is again a weak solution to a limit system.

Blow-up solutions of nonlinear elliptic equations in R^n with critical exponent

For an integer $n \ge 3$ and any positive number $ε$ we establish the existence of smooth functions K on $R^n \setminus \{0 \}$ with $|K - 1| \le ε$, such that the equation $Δu + n (n - 2) K u^{{n + 2}\over {n - 2}} = 0$ in $R^n \setminus \{0 \}$ has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some cases K can be extended as a Lipschitz function on ${\R}^n.$ These provide counter-examples to a conjecture of C.-S. Lin when n > 4, and Taliaferro's conjecture.

Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

In this paper we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.

Compactness and existence results for degenerate critical elliptic equations

This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type. By means of blow-up analysis techniques, we prove an a-priori estimate in a weighted space of continuous functions. From this compactness result, the existence of a solution to our problem is proved by exploiting the homotopy invariance of the Leray-Schauder degree.

Multiplicity results for $(p,\, q)$ fractional elliptic equations involving critical nonlinearities

In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_Ω(Ω)$ nonnegative solutions.

An abstract critical point theorem with applications to elliptic problems with combined nonlinearities

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.

Existence of positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in $\mathbb{R}^{2}$

In this paper, we use variational methods to prove the existence of positive solution for the following class of elliptic equation $$ -ε^{2}Δ{u}+V(z)u=f(u) \,\,\, \mbox{in} \,\,\, \mathbb{R}^{2}, $$ where $ε>0$ is a positive parameter, $V$ is a saddle-like potential and $f$ has an exponential critical growth.

具有临界指数增长的椭圆问题的多重性结果

Multiplicity results for elliptic problems with critical exponential growth

本文研究具有临界指数增长的椭圆问题，利用抽象临界点定理证明当参数$\mu$充分大并趋于无穷时，问题存在任意多个解且解的数量趋于无穷。

混合局部与非局部半线性椭圆方程，含强奇异和临界Choquard非线性项

Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity

本文研究混合局部与非局部椭圆方程，结合奇异与临界非线性，通过变分方法证明正解的存在性与多重性。

齐次分数阶 Sobolev 空间中的非局部临界椭圆方程

Nonlocal critical elliptic equations in homogeneous fractional Sobolev spaces

本文基于 ${\mathbb Z}_2$-上同调指标的抽象临界点定理、分数阶 $p$-Laplacian 方程的正则性结果与紧嵌入，证明了齐次分数阶 Sobolev 空间中非局部临界增长椭圆方程的多重性结果。

Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities

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.

Critical blowup exponent to a class of semilinear elliptic equations with constraints in higher dimension - local properties

We study a class of semilinear elliptic equations with constraints in higher dimension. It is known that several mathematical structures of the problem are closed to those of the Liouville equation in dimension two. In this paper, we establish a classification of entire solutions, the $\sup + \inf$ type inequality and the quantized blowup mechanism.

Infinitely many sign-changing solutions for a class of elliptic problem with exponential critical growth

In this work we prove the existence of infinitely many nonradial solutions that change signal to the problem $-Δu=f(u)$ in $B$ with $u=0$ on $\partial B$, where $B$ is the unit ball in $\mathbb{R}^2$ and $f$ is a continuous and odd function with exponential critical growth.

On a nonlinear fourth order elliptic equation involving the Sobolev critical exponent

In this paper we use an algebraic topological argument due to Bahri and Coron to show how the topology of the domain influences the existence of positive solutions of some fourth order elliptic equation involving the critical Sobolev exponent.

Sign-changing solutions to elliptic second order equations: glueing a peak to a degenerate critical manifold

We construct blowing-up sign-changing solutions to some nonlinear critical equations by glueing a standard bubble to a degenerate function. We develop a method based on analyticity to perform the glueing when the critical manifold of solutions is degenerate and no Bianchi--Egnell type condition holds.

Large critical exponents for some second order uniformly elliptic operators

In this paper we investigate the critical exponents of two families of Pucci's extremal operators. The notion of critical exponent that we have chosen for these fully nonlinear operators whihc are not variational is that of threshold betweeen existence and nonexistence of the solutions for semilinear equations with pure power nonlinearities. Interesting new exponents appear in this context.

Ground state solutions for a semilinear elliptic problem with critical-subcritical growth

In this work, we study the of positive ground state solution for the semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -Δu=u^{p(x)-1},\quad u>0 & \mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\ u\in D_{0}^{1,2}(G), & \end{array} \right. $$ where $G$ is either $\mathbb{R}^{N}$ or a bounded domain, and $p:G\rightarrow \mathbb{R}$ is a continuous function assuming critical and subcritical values.

Concave-convex effects for critical quasilinear elliptic problems

It is established existence, multiplicity and asymptotic behavior of positive solutions for a quasilinear elliptic problem driven by the $Φ$-Laplacian operator. One of these solutions is obtained as ground state solution by applying the well known Nehari method. The semilinear term in the quasilinear equation is a concave-convex function which presents a critical behavior at infinity. The concentration compactness principle is used in order to recover the compactness required in variational methods.

Positive clusters for smooth perturbations of a critical elliptic equation in dimensions four and five

We construct clustering positive solutions for a perturbed critical elliptic equation on a closed manifold of dimension $n=4,5$. Such a construction is already available in the literature in dimensions $n\ge 6$ (see for instance [8,12,27,29,33]) and not possible in dimension $3$ by [25]. This also provides new patterns for the Lin--Ni [21] problem on closed manifolds and completes results by Brézis and Li [6] about this problem.

On local compactness in quasilinear elliptic problems

One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work \cite{BN}, Brézis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relativ…

Positive solutions of semipositone elliptic problems with critical Trudinger-Moser nonlinearities

We prove the existence of a positive solution to a semipositone $N$-Laplacian problem with a critical Trudinger-Moser nonlinearity. The proof is based on obtaining uniform $C^{1,α}$ a priori estimates via a compactness argument. Our result is new even in the semilinear case $N = 2$, and our arguments can easily be adapted to obtain positive solutions of more general semipositone problems with critical Trudinger-Moser nonlinearities.

有界区域中带临界源的Orlicz-Sobolev空间中的椭圆方程组

Elliptic systems in Orlicz-Sobolev spaces with critical sources in bounded domains

该研究在Orlicz-Sobolev空间框架下，结合无Palais-Smale条件的山路定理与Lions第二集中紧性原理，证明了带临界非线性项的一般椭圆方程组非平凡解的存在性，可处理非多项式增长的非线性项。

具有临界指数和对数项的椭圆方程组的正解

Positive solution for an elliptic system with critical exponent and logarithmic terms

针对含临界指数与对数项的耦合椭圆方程组，本文研究其正解存在性与不存在性，并指出对数项要求超越 Chen 和 Zou 既有方法的新技术。

Moser iteration applied to elliptic equations with critical growth on the boundary

This paper deals with boundedness results for weak solutions of an elliptic equation where the functions are Carathéodory functions satisfying certain $p$-structure conditions that have critical growth even on the boundary. Based on a modified version of the Moser iteration we are able to prove that every weak solution of our problem is bounded up to the boundary. Under some additional assumptions this leads directly to $C^{1,α}$-regularity for weak solutions of the problem.

具有齐次临界非线性的各向异性拟线性椭圆系统

Anisotropic Quasilinear Elliptic Systems with Homogeneous Critical Nonlinearities

研究各向异性 p-拉普拉斯拟线性椭圆系统，给出保证非平凡解存在与不存在的条件，涉及临界 Sobolev 增长。

Asymptotic Behaviors of Solutions to quasilinear elliptic Equations with critical Sobolev growth and Hardy potential

Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \[-Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\,x\in \mathbb{R}^{N},\] where $1<p<N, 0\leqμ<\left({(N-p)}/{p}\right)^{p}$ and $Q\in L^{\infty}(\mathbb{R}^{N})$.

关于具有临界指数增长的拟单调椭圆系统的存在性结果

Existence results for quasimonotone elliptic systems with growth up to critical exponents

本文研究了具有临界指数增长的拟单调椭圆系统弱解的存在性，采用非线性偏微分方程的分析方法。

Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient

We prove non-uniqueness and study the behaviour of viscosity solutions of a class of uniformly elliptic fully nonlinear equations of Hamilton-Jacobi-Bellman-Isaacs type, with quadratic growth in the gradient. The crucial a priori bound for the solutions is proved through an argument which uses a boundary growth lemma, and consequences such as boundary "half"-Harnack inequalities, which are of independent interest. Our results are new even for linear equations.

Global existence of solutions to a parabolic-elliptic chemotaxis system with critical degenerate di ffusion

This paper is devoted to the analysis of non-negative solutions for a degenerate parabolic-elliptic Patlak-Keller-Segel system with critical nonlinear diffusion in a bounded domain with homogeneous Neumann boundary conditions. Our aim is to prove the existence of a global weak solution under a smallness condition on the mass of the initial data, there by completing previous results on nite blow-up for large masses. Under some higher regularity condition on solutions, the uniqueness of solutions is proved by using a classical duality technique.

临界双曲线上带超线性项的椭圆系统的正解

Positive solutions of elliptic systems with superlinear terms on the critical hyperbola

本文研究带Dirichlet边界条件的略次临界椭圆系统，利用Orlicz空间上的对偶方法，证明了山隘型正解的存在性。

Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent

We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas, and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [11] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.

Liouville properties and critical value of fully nonlinear elliptic operators

We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.

Very degenerate elliptic equations under almost critical Sobolev regularity

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.

Existence of bound and ground states for an elliptic system with double criticality

We study the existence of bound and ground states for a class of nonlinear elliptic systems in $\mathbb{R}^N$. These equations involve critical power nonlinearities and Hardy-type singular potentials, coupled by a term containing up to critical powers. More precisely, we find ground states either the positive coupling parameter $ν$ is large or $ν$ is small under suitable assumptions on the other parameters of the problem. Furthermore, bound states are found as Mountain-Pass-type critical points of the underlying functional constrained on the Nehari manifold. Our variational approach improves s…

关于涉及局部和非局部算子差的临界椭圆问题的多重解结果

A multiplicity result for critical elliptic problems involving differences of local and nonlocal operators

本文研究了涉及局部与非局部算子差的临界椭圆问题，证明了对于足够小的参数值存在两个非平凡弱解。

Critical fractional elliptic equations with exponential growth without Ambrosetti-Rabinowitz type condition

In this paper we establish, using variational methods combined with the Moser-Trudinger inequality, existence and multiplicity of weak solutions for a class of critical fractional elliptic equations with exponential growth without a Ambrosetti-Rabinowitz-type condition. The interaction of the nonlinearities with the spectrum of the fractional operator will used to study the existence and multiplicity of solutions. The main technical result proves that a local minimum in $C_{s}^0(\overlineΩ)$ is also a local minimum in $W^{s,p}_0$ for nonlinearities with exponential growth.

Removable singularity of positive solutions for a critical elliptic system with isolated singularity

We study qualitative properties of positive singular solutions to a two-coupled elliptic system with critical exponents. This system is related to coupled nonlinear Schrodinger equations with critical exponents for nonlinear optics and Bose-Einstein condensates. We prove a sharp result on the removability of the same isolated singularity for both two components of the solutions. We also prove the nonexistence of positive solutions with one component bounded near the singularity and the other component unbounded near the singularity. These results will be applied in a subsequent work where the …

Pairs of nontrivial solutions to concave-linear-convex type elliptic problems

We obtain a pair of nontrivial solutions for a class of concave-linear-convex type elliptic problems that are either critical or subcritical. The solutions we find are neither local minimizers nor of mountain pass type in general. They are higher critical points in the sense that they each have a higher critical group that is nontrivial. This fact is crucial for showing that our solutions are nontrivial. We also prove some intermediate results of independent interest on the localization and homotopy invariance of critical groups of functionals involving critical Sobolev exponents.