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Register a new userA complete study of the lack of compactness and existence results of a Fractional Nirenberg Equation via a flatness hypothesis: Part I
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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.
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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
The paper provides an extension, to fractional order Sobolev spaces, of the classical result of Murat and Brezis which states that the positive cone of elements in $H^{-1}(Omega)$ compactly embeds in $W^{-1,q}(Omega)$, for every $q < 2$ and for any open and bounded set $Omega$ with Lipschitz boundary. In particular, our proof contains the classical result. Several new analysis tools are developed during the course of the proof to our main result which are of wider interest. Subsequently, we apply our results to the convergence of convex sets and establish a fractional version of the Mosco convergence result of Boccardo and Murat. We conclude with an application of this result to quasi-variational inequalities.
Let $Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $Sigma$, $rho$ and $alpha$ real numbers. In this paper, we study a generalized mean field equation begin{align*} -Delta u=rholeft(dfrac{he^u}{int_Sigma he^u}-dfrac{1}{mathrm{Area}left(Sigmaright)}right)+alphaleft(u-fint_{Sigma}uright), end{align*} where $Delta$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $rhoin (8kpi, 8(k+1)pi)$ for some non-negative integer number $kin mathbb{N}$ and $alpha otinmathrm{Spec}left(-Deltaright)setminusset{0}$. Then we obtain existence results for $alpha<lambda_1left(Sigmaright)$ by using the Leray-Schauder degree theory and the minimax method, where $lambda_1left(Sigmaright)$ is the first positive eigenvalue for $-Delta$.
There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the Riesz characteristic, which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-continuous density function. Two theorems establishing the strong uniqueness of tangents (which means every tangent is a Riesz kernel) are proved. They cover all O(n)-invariant convex cone equations and their complex and quaternionic analogues, with the exception of the homogeneous Monge-Amp`ere equations, where uniqueness fails. They also cover a large class of geometrically defined subequations which includes those coming from calibrations. A discreteness result for the sets where the density is $geq c > 0$ is also established in any case where strong uniqueness holds. A further result (which is sharp) asserts the Holder continuity of subsolutions when the Riesz characteristic p satisfies $1 leq p < 2$. Many explicit examples are examined. The second part of this paper is devoted to the geometric cases. A Homogeneity Theorem and a Second Strong Uniqueness Theorem are proved, and the tangents in the Monge-Amp`ere case are completely classified.
Let $Omegasubsetmathbb{R}^n$ be a $C^2$ bounded domain and $chi>0$ be a constant. We will prove the existence of constants $lambda_Ngelambda_N^{ast}gelambda^{ast}(1+chiint_{Omega}frac{dx}{1-w_{ast}})^2$ for the nonlocal MEMS equation $-Delta v=lam/(1-v)^2(1+chiint_{Omega}1/(1-v)dx)^2$ in $Omega$, $v=0$ on $1Omega$, such that a solution exists for any $0lelambda<lambda_N^{ast}$ and no solution exists for any $lambda>lambda_N$ where $lambda^{ast}$ is the pull-in voltage and $w_{ast}$ is the limit of the minimal solution of $-Delta v=lam/(1-v)^2$ in $Omega$ with $v=0$ on $1Omega$ as $lambda earrow lambda^{ast}$. We will prove the existence, uniqueness and asymptotic behaviour of the global solution of the corresponding parabolic nonlocal MEMS equation under various boundedness conditions on $lambda$. We also obtain the quenching behaviour of the solution of the parabolic nonlocal MEMS equation when $lambda$ is large.
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