We prove the non-degeneracy for the critical Lane--Emden system $$ -ΔU = V^p,\quad -ΔV = U^q,\quad U, V > 0 \quad \text{in } \mathbb{R}^N $$ for all $N \ge 3$ and $p,q > 0$ such that $\frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}$. We show that all solutions to the linearized system around a ground state must arise from the symmetries of the critical Lane-Emden system provided that they belong to the corresponding energy space or they decay to 0 uniformly as the point tends to infinity.
We establish a nonlinear instability of the Euler-Poisson system for polytropic gases whose adiabatic exponents take value in $6/5<γ<4/3$ around the Lane-Emden equilibrium star configurations.
We deal with very weak positive supersolutions to the Hénon-Lane-Emden system on neighborhoods of the origin. In our main theorem we prove a sharp nonexistence result.
In this paper, we present a necessary and sufficient condition to the Lane-Emden conjecture. This condition is an energy type of integral estimate on solutions to subcritical Lane-Emden system. To approach the long standing and interesting conjecture, we believe that one plausible path is to refocus on establishing this energy type estimate.
We discuss on existence, nonexistence and uniqueness of positive viscosity solutions for Lane-Emden systems involving the fractional Laplacian on bounded domains. As a byproduct, we obtain the critical hyperbole associated to the these systems.
We consider the Lane-Emden conjecture which states that there is no non-trivial non-negative solution for the Lane-Emden system whenever the pair of exponents is subcritical. By Sobolev embeddings on $S^{N-1}$ and scale invariance of the solutions, we show this conjecture holds in a new region. Our methods can also be used to prove the Lane-Emden conjecture in space dimension $N\leq 4$, that is to give a different proof of the main result of Souplet in Adv. Math. 2009.
We use variational methods to study the existence of a principal eigenvalue for the non-anticoercive Hénon-Lane-Emden system on a bounded domain. Then we provide a detailed insight into the problem in the linear case.
We prove that 0 the only classical solution of the Lane-Emden equation in the half-space which is stable outside a compact set. We also consider weak solutions and the case of general cones.
In this short note, we prove a sharp quantization for positive solutions of Lane-Emden problems in a bounded planar domain. This result has been conjectured by De Marchis, Ianni and Pacella [6, Remark 1.2].
We consider the Lane-Emden system $-Δu = v^p$, $-Δv= u^θ$ in $\mathbb{R}^N$, and we prove the nonexistence of smooth positive solutions which are stable outside a compact set, for any $p, θ> 0$ under the Sobolev hyperbola.
We are interested in studying positive solutions of Lane-Emden equation $$ -Δu= V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\}, $$ where $V$ is a nonhomogeneous potential satisfying some extra hypotheses. We construct a sequence of fast decaying solutions.
We study stable positive radially symmetric solutions for the Lane-Emden system $-Δu=v^p$ in $\R^N$, $-Δv=u^q$ in $\R^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.
In this paper we are concerned with the least energy solutions to the Lane-Emden problem driven by an anisotropic operator, so-called the Finsler $N$-Laplacian, on a bounded domain in $\mathbb{R}^N$. We prove several asymptotic formulae as the nonlinear exponent gets large.
We classify the stable solutions (positive or sign-changing, radial or not) to the following nonlocal Lane-Emden equation: $(-Δ)^s u=|u|^{p-1}u$ in $\mathbb{R}^n$ for $2<s<3$.
In this paper, we compute the Joseph-Lundgren exponent for the quadharmonic Lane-Emden equation, derive a monotonicity formula and classify the finite Morse index solution to the following quadharmonic Lane-Emden equation: \noindent \begin{equation}\nonumber Δ^4 u=|u|^{p-1}u\;\;\;\;\hbox{in}\;\;\;\;\; \R^n. \end{equation} As a byproduct, we also get a monotonicity formula for the quadharmonic maps $ Δ^4 u=0$.
We study qualitative properties of solutions to the fractional Lane-Emden-Fowler equations with slightly subcritical exponents where the associated fractional Laplacian is defined in terms of either the spectra of Dirichlet Laplacian or the integral representation. As a consequence, we classify the asymptotic behavior of all finite energy solutions. Our method also provides a simple and unified approach to deal with the classical (local) Lane-Emden-Fowler equation for any dimension greater than 2.
We prove that positive solutions of the superlinear Lane-Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded. In addition, the boundedness may fail if the exponents are not comparable.
We prove that on a smooth bounded set, the positive least energy solution of the Lane-Emden equation with sublinear power is isolated. As a corollary, we obtain that the first $q-$eigenvalue of the Dirichlet-Laplacian is not an accumulation point of the $q-$spectrum, on a smooth bounded set. Our results extend to a suitable class of Lipschitz domains, as well.
In this paper we prove the existence of continua of nonradial solutions for the Lane-Emden equation. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane Emden problem has a very rich and complex structure.
In this paper, we are concerned with the critical order Lane-Emden-Hardy equations \begin{equation*} (-Δ)^{\frac{n}{2}}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq4$ is even, $0\leq a<n$ and $1<p<+\infty$. We prove Liouville theorem for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv0$. Our result seems to be the first Liouville theorem on the critical order equations in higher dimensions ($n\geq3$).
We derive monotonicity formulae for solutions of the fractional Hénon-Lane-Emden equation \begin{equation*} (-Δ)^{s} u=|x|^a |u|^{p-1} u \ \ \ \text{in } \ \ \mathbb{R}^n, \end{equation*} when $0<s<2$, $a>0$ and $p>1$. Then, we apply these formulae to classify stable solutions of the above equation.
In this paper, we first establish decay estimates for the fractional and higher-order fractional Hénon-Lane-Emden systems by using a nonlocal average and integral estimates, which deduce a result of non-existence. Next, we apply the method of scaling spheres introduced in \cite{DQ2} to derive a Liouville type theorem. We also construct an interesting example on super $\fracα{2}$-harmonic functions (Proposition 1.2).
We classify finite Morse index solutions of the fractional Lane-Emden equation $(-Δ)^{s} u=|u|^{p-1} u \ \ \ \mathbb{R}^n $ for $1<s<2$. For the local case, $s=1$ and $s=2$ this classification was done by Farina in [10] and Davila, Dupaigne, Wang and Wei in [8], respectively. Moreover, for the nonlocal case, $0<s<1$, finite Morse index solutions are classified by Davila, Dupaigne and Wei in [7].
We study the Lane-Emden system involving the logarithmic Laplacian: $$ \begin{cases} \ \mathcal{L}_Δu(x)=v^{p}(x) ,& x\in\mathbb{R}^{n},\\ \ \mathcal{L}_Δv(x)=u^{q}(x) ,& x\in\mathbb{R}^{n}, \end{cases} $$ where $p,q>1$ and $\mathcal{L}_Δ$ denotes the Logarithmic Laplacian arising as a formal derivative $\partial_s|_{s=0}(-Δ)^s$ of fractional Laplacians at $s=0.$ By using a direct method of moving planes for the logarithmic Laplacian, we obtain the symmetry and monotonicity of the positive solutions to the Lane-Emden system. We also establish some key ingredients needed in order to apply the m…
For any smoothly bounded domain $Ω\subset\mathbb R^n$, $n\geq 3$, and any exponent $p>2^*=2n/(n-2)$ we study the Lane-Emden heat flow $u_t-Δu = |u|^{p-2}u$ on $Ω\times]0,\infty[$ and establish local and global well-posedness results for the initial value problem with suitably small initial data $u\big|_{t=0}=u_0$ in the Morrey space $L^{2,λ}(Ω)$, where $λ=4/(p-2)$. We contrast our results with results on instantaneous complete blow-up of the flow for certain large data in this space, similar to ill-posedness results of Galaktionov-Vazquez for the Lane-Emden flow on $\mathbb R^n$.
We establish the existence of finitely many sign-changing solutions to the Lane-Emden system $$-Δu=|v|^{q-2}v,\quad -Δv=|u|^{p-2}u \quad \text{ in }\mathbb{R}^N, \ \ N\geq 4,$$ where the exponents $p$ and $q$ lie on the critical hyperbola $\frac{1}{p}+\frac{1}{q}=\frac{N-2}{N}$. These solutions are nonradial and arise as limit profiles of symmetric sign-changing minimizing sequences for a critical higher-order problem in a bounded domain.
We consider the Lane-Emden Dirichlet problem \begin{equation}\tag{1} \left\{\begin{array}{lr}-Δu= |u|^{p-1}u\qquad \mbox{ in }Ωu=0\qquad\qquad\qquad\mbox{ on }\partial Ω\end{array}\right. \end{equation} when $p>1$ and $Ω\subset\mathbb R^2$ is a smooth bounded domain. The aim of the paper is to survey some recent results on the asymptotic behavior of solutions of (1) as the exponent $p\rightarrow \infty $.
We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality.
In this paper, we prove Liouville type theorems for stable solutions to the weighted fractional Lane-Emden system \begin{align*} (-Δ)^s u = h(x)v^p,\quad (-Δ)^s v= h(x)u^q, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N, \end{align*} where $1<q\leq p$ and $h$ is a positive continuous function in $\mathbb{R}^N$ satisfying $\displaystyle{\liminf_{|x|\to \infty}}\frac{h(x)}{|x|^\ell} > 0$ with $\ell > 0.$ Our results generalize the results established in \cite{HHM16} for the Laplacian case (correspond to $s=1$) and improve the previous work \cite{TuanHoang21}. As a consequence, we prove classification…
In this paper, we study the existence of uniform a priori estimates for positive solutions to Navier problems of higher order Lane-Emden equations \begin{equation*} (-Δ)^{m}u(x)=u^{p}(x), \qquad \,\, x\inΩ\end{equation*} for all large exponents $p$, where $Ω\subset\mathbb{R}^{n}$ is a star-shaped or strictly convex bounded domain with $C^{2m-2}$ boundary, $n\geq4$ and $2\leq m\leq\frac{n}{2}$. Our results extend those of previous authors for second order $m=1$ to general higher order cases $m\geq2$.
In this paper, we are concerned with the uniqueness result for non-negative solutions of the higher-order Lane-Emden equations involving the GJMS operators on $\mathbb{S}^n$. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in $\mathbb{S}^n$, we first employ the Mobius transform between $\mathbb{S}^n$ and $\mathbb{R}^n$, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on $\mathbb{S}^n$ is equivalent to some integral equation in $\mathbb{R}^n$. Then …
In this paper, we are concerned with the non-critical higher order Lane-Emden-Hardy equations \begin{equation*} (-Δ)^{m}u(x)=\frac{u^{p}(x)}{|x|^{a}} \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} with $n\geq3$, $1\leq m<\frac{n}{2}$, $0\leq a<2m$, $1<p<\frac{n+2m-2a}{n-2m}$ if $0\leq a<2$, and $1<p<\infty$ if $2\leq a<2m$. We prove Liouville theorems for nonnegative classical solutions to the above Lane-Emden-Hardy equations (Theorem \ref{Thm0}), that is, the unique nonnegative solution is $u\equiv0$. As an application, we derive a priori estimates and existence of p…
This paper proves the nonlinear asymptotic stability of the Lane-Emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant $γ$ lies in the stability range $(4/3, 2)$. It is shown that for small perturbations of a Lane-Emden solution with same mass, there exists a unique global (in time) strong solution to the vacuum free boundary problem of the compressible Navier-Stokes-Poisson system with spherical symmetry for viscous stars, and the solution captures the precise physical behavior that the sound speed is $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous …
We examine the general weighted Lane-Emden system \begin{align*} -Δu = ρ(x)v^p,\quad -Δv= ρ(x)u^θ, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N \end{align*} where $1<p\leqθ$ and $ρ: \mathbb{R}^N\rightarrow \mathbb{R}$ is a radial continuous function satisfying $ρ(x)\geq A(1+|x|^2)^{\fracα{2}}$ in $\mathbb{R}^N$ for some $α\geq 0$ and $A>0$. We prove some Liouville type results for stable solution and improve the previous works \cite{co, Fa, HU}. In particular, we establish a new comparison property (see Proposition 1.1 below) which is crucial to handle the case $1 < p \leq \frac{4}{3}$. Our resul…
In this paper we exploit the phenomenon of two principal half eigenvalues in the context of fully nonlinear Lane-Emden type systems with possibly unbounded coefficients and weights. We show that this gives rise to the existence of two principal spectral curves on the plane. We also construct a possible third spectral curve related to a second eigenvalue and an anti-maximum principle, which are novelties even for Lane-Emden systems involving linear operators. As applications, we derive a maximum principle in small domains for these systems, as well as existence and uniqueness of positive soluti…
We study Liouville-type theorem for polyharmonic Hénon-Lane-Emden system $(-Δ)^mu=|x|^av^p,\; (-Δ)^mv=|x|^bu^q$ when $m,p,q\geq 1, pq\ne 1$, and $a,b\geq 0$. It is a natural conjecture that the nonexistence of positive solutions should be true if and only if $(N+a)/(p+1)+$ $(N+b)/(q+1)>N-2m$. It is shown by Fazly [6] that the conjecture holds for radial solutions in all dimensions and for classical solutions in dimension $N\leq 2m+1$. We here give some partial results in dimension $N\geq 2m+2$.
We study existence, nonexistence, and uniqueness of positive radial solutions for a class of nonlinear systems driven by Pucci extremal operators under a Lane-Emden coupling configuration. Our results are based on the analysis of the associated quadratic dynamical system and energy methods. For both regular and exterior domain radial solutions we obtain new regions of existence and nonexistence. Besides, we show an exclusion principle for regular solutions, either in $\mathbb{R}^N$ or in a ball, by exploiting the uniqueness of trajectories produced by the flow. In particular, for the standard …
We are concerned with the Lane-Emden problem \begin{equation*} \begin{cases} -Δu=u^{p} &{\text{in}~Ω},\\[0.5mm] u>0 &{\text{in}~Ω},\\[0.5mm] u=0 &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $Ω\subset \mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large. Improving some known asymptotic estimates on the solutions, we prove the non-degeneracy and local uniqueness of the multi-spikes positive solutions for general domains. Our methods mainly use ODE's theory, various local Pohozaev identities, blow-up analysis and the properties of Green's function.
We consider ground state solutions of the critical Lane-Emden system \[\begin{cases} -Δu = v^p &\text{in } \mathbb{R}^n,\\ -Δv = u^q &\text{in } \mathbb{R}^n,\\ u,v >0\ &\text{in } \mathbb{R}^n, \end{cases}\] where $n \ge 3$ and $p,q>0$ and $(p,q)$ belongs to the critical hyperbola $\frac{1}{p+1} + \frac{1}{q+1} = \frac{n-2}{n}.$ We prove that they are non-degenerate when either $(p,q)$ is close to $(1,{n+4\over n-4})$ (if $n\ge5$) or $(p,q)$ is close to $({n+2\over n-2},{n+2\over n-2})$ (if $n\ge3$).
We consider the semilinear Lane Emden problem in a smooth bounded simply connected domain in the plane, invariant by the action of a finite symmetry group G. We show that if the orbit of each point in the domain, under the action of the group G, has cardinality greater than or equal to four then, for p sufficiently large, there exists a sign changing solution of the problem with two nodal regions whose nodal line does not touch the boundary of the domain. This result is proved as a consequence of an analogous result for the associated parabolic problem.
The Lane-Emden system is written as \begin{equation*} \begin{cases} -Δu = v^p &\text{in } Ω,\\ -Δv = u^q &\text{in } Ω,\\ u, v > 0 &\text{in } Ω,\\ u = v = 0 &\text{on } \partial Ω\end{cases} \end{equation*} where $Ω$ is a smooth bounded domain in the Euclidean space $\mathbb{R}^n$ for $n \ge 3$ and $0< p< q <\infty$. The asymptotic behavior of least energy solutions near the critical hyperbola was studied by Guerra \cite{G} when $p \geq 1$ and the domain is convex. In this paper, we cover all the remaining cases $p < 1$ and extend the results to any smooth bounded domain.
We consider the semilinear Lane-Emden problem: \begin{equation}\label{problemAbstract}\left\{\begin{array}{lr}-Δu= |u|^{p-1}u\qquad \mbox{ in }B u=0\qquad\qquad\qquad\mbox{ on }\partial B \end{array}\right.\tag{$\mathcal E_p$} \end{equation} where $B$ is the unit ball of $\mathbb R^N$, $N\geq3$, centered at the origin and $1<p<p_S$, $p_S=\frac{N+2}{N-2}$. We prove that for any radial solution $u_p$ of \eqref{problemAbstract} with $m$ nodal domains its Morse index $\mathsf{m}(u_p)$ is given by the formula \[\mathsf{m}(u_p)=m+N(m-1)\] if $p$ is sufficiently close to $p_S$.
We consider the semilinear Lane-Emden problem in a smooth bounded domain of the plane. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions as the exponent p of the nonlinearity goes to infinity. Among other results we show, under some symmetry assumptions on the domain, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p goes to infinity, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville problem in the plane.
