相关论文: Sharp subcritical Sobolev inequalities and uniquen…

We present a novel, high-order, efficient, and exponentially convergent shifted Gegenbauer integral pseudospectral method (SGIPSM) to solve numerically Lane-Emden equations provided with some mixed Neumann and Robin boundary conditions. The…

In this paper, we study the following high order elliptic equation involving the GJMS operator: \begin{align*} \alpha P_{\mathbb{S}^n}v_{\alpha}+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_{\alpha}}. \end{align*} We establish that if…

We study positive solutions of the superlinear Lane-Emden inequality \(-\Delta u\ge \sigma u^q\), \(q>1\), on infinite locally finite weighted graphs and connected domains of such graphs. We first prove that solvability is equivalent to the…

Let $N\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation $\Delta u + u^p = 0$ in $\mathbb{R}^N$ and recall the classical Liouville type theorem: if $u$ is a non-negative classical solution of the Lane-Emden equation,…

In this paper we analyse the Lane-Emden system \begin{equation} \left\{ \begin{alignedat}{3} -\Delta u = & \, \frac{\lambda f(x)}{(1-v)^2} & \quad \text{in} & \quad\Omega\\ -\Delta v = & \, \frac{\mu g(x)}{(1-u)^2} & \quad \text{in} &…

This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the…

In this paper we consider lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain $\Omega \subset \mathbb{R}^N$, with $N \geq3$, inspired by the classical results of Brezis and Nirenberg…

This paper presents high-order numerical methods for solving boundary value problems associated with the Lane-Emden equation, which frequently arises in astrophysics and various nonlinear models. A major challenge in studying this equation…

We prove existence results for the Lane-Emden type system \[ \begin{cases} \begin{aligned} (-\Delta)^{\alpha} u=\left| v \right|^q \\ (-\Delta)^{\beta} v= \left| u \right|^p \end{aligned} \text{ in } B_1 \subset \mathbb{R}^N \\…

A new analytic approximate technique for addressing nonlinear problems, namely the optimal perturbation iteration method, is introduced and implemented to singular initial value Lane-Emden type problems to test the effectiveness and…

In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov-Serrin and a…

We study the Lane-Emden system involving the logarithmic Laplacian: $$ \begin{cases} \ \mathcal{L}_{\Delta}u(x)=v^{p}(x) ,& x\in\mathbb{R}^{n},\\ \ \mathcal{L}_{\Delta}v(x)=u^{q}(x) ,& x\in\mathbb{R}^{n}, \end{cases} $$ where $p,q>1$ and…

In this paper, we establish uniform a priori estimates for positive solutions to the (higher) critical order superlinear Lane-Emden system in bounded domains with Navier boundary conditions in arbitrary dimensions $n\geq3$. First, we prove…

We study the Lane-Emden system $$\begin{cases} -\Delta u=v^p,\quad u>0,\quad\text{in}~\Omega, -\Delta v=u^q,\quad v>0,\quad\text{in}~\Omega, u=v=0,\quad\text{on}~\partial\Omega, \end{cases}$$ where $\Omega\subset\mathbb{R}^2$ is a smooth…

In this paper, we decide to compare two new approaches based on Rational and Exponential Bessel functions (RBs and EBs) to solve several well-known class of Lane-Emden type models. The problems, which define in some models of non-Newtonian…

The Newton, Gauss--Newton and Levenberg--Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some…

In this paper we establish the reversed sharp Hardy-Littlewood-Sobolev (HLS for short) inequality on the upper half space and obtain a new HLS type integral inequality on the upper half space (extending an inequality found by Hang, Wang and…

We prove a comparison principle for positive supersolutions and subsolutions to the Lane-Emden equation for the $p-$Laplacian, with subhomogeneous power in the right-hand side. The proof uses variational tools and the result applies with no…

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in…

We focus on the problems of existence and non-existence of positive solutions for the Sobolev-subcritical Lane-Emden equation on certain Riemannian manifolds (mainly models) with asymptotically negative curvature, which, from the viewpoint…