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Register a new userExistence Results for a critical fractional equation
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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
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This paper is devoted to study the existence and multiplicity solutions for the nonlinear Schrodinger-Poisson systems involving fractional Laplacian operator: begin{equation}label{eq*} left{ aligned &(-Delta)^{s} u+V(x)u+ phi u=f(x,u), quad &text{in }mathbb{R}^3, &(-Delta)^{t} phi=u^2, quad &text{in }mathbb{R}^3, endaligned right. end{equation} where $(-Delta)^{alpha}$ stands for the fractional Laplacian of order $alphain (0,,,1)$. Under certain assumptions on $V$ and $f$, we obtain infinitely many high energy solutions for eqref{eq*} without assuming the Ambrosetti-Rabinowitz condition by using the fountain theorem.
We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ partial_t u + (-Delta)^{ 1/2} u = | abla u|^p, quad x in mathbb R^N, t > 0, qquad u(x,0) = u_0(x) , quad x in mathbb R^N, $$ where $p > 1$ and $u_0 in B^1_{r,1}(mathbb R^N) cap B^1_{infty,1} (mathbb R^N)$ with $r in [1,infty]$. We show that for sufficiently small $u_0 in dot B^1_{infty,1}(mathbb R^N)$, there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel.
We study the existence of nontrivial solutions for a nonlinear fractional elliptic equation in presence of logarithmic and critical exponential nonlinearities. This problem extends [5] to fractional $N/s$-Laplacian equations with logarithmic nonlinearity. We overcome the lack of compactness due to the critical exponential nonlinearity by using the fractional Trudinger-Moser inequality. The existence result is established via critical point theory.
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 $-Delta$.
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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