Let $p$ and $q$ be locally Hölder functions in $\RR^N$, $p>0$ and $q\geq 0$. We study the Emden-Fowler equation $-Δu+ q(x)|\nabla u|^a=p(x)u^{-γ}$ in $\RR^N$, where $a$ and $γ$ are positive numbers. Our main result establishes that the above equation has a unique positive solutions decaying to zero at infinity. Our proof is elementary and it combines the maximum principle for elliptic equations with a theorem of Crandall, Rabinowitz and Tartar.
For one spatial variable, a new kind of nonlinear wave equation for Emden-Fowler type is considered with boundary value null and initial values. Under certain conditions on the initial data and the exponent p, we exhibit that the visco-elastic term leads our problem to be dissipative and the global solutions still non-exist in L2 at given finite time.
A generalization of the Emden-Fowler equation is presented and its solutions are investigated. This paper is devoted to asymptotic behavior of its solutions. The procedure is entirely based on a previous paper by the author.
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 study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-Δ_g u=|u|^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space $\mathbb H^n$ as well as manifolds with sectional curvatures unbounded below. Sign properties and asymptotic behavior of solutions are influenced by the critical Sobolev exponent while the so-called Joseph-Lundgren exponent is involved in the stability of solutions.
We study the Emden-Fowler equation $-Δu=|u|^{p-1}u$ on the hyperbolic space ${\mathbb H}^n$. We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is $p=(n+2)/(n-2)$ as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers \cite{mancini, bhakta} consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy…
We are concerned with singular elliptic equations of the form $-Δu= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with sublinear growth and $g$ is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle.
We consider a class of weighted Emden-Fowler equations \begin{equation} \tag{$\mathcal P_α$} \label{eqab} \left\{\begin{array}{ll} -Δu=V_α (x) \, u^p & \text{in} \,\,B,\\ u>0 & \text{in} \,\,B,\\ u=0 & \text{on}\,\,\partial B, \end{array}\right. \end{equation} posed on the unit ball $B=B(0,1)\subset \mathbb R^N$, $N \geq1$. We prove that symmetry breaking occurs for the groundstate solutions as the parameter $α\rightarrow \infty.$ The above problem reads as a possibly large perturbation of the classical Hénon equation. We consider a radial function $V_α$ having a spherical shell of zeroes at $…
Let $\mathbb{S}^N\subset\mathbb{R}^{N+1}$, $N\ge 3$, be the unit sphere, and let $S_Θ\subset\mathbb{S}^N$ be a geodesic ball with geodesic radius $Θ\in(0,π)$. We study the bifurcation diagram $\{(Θ,\left\|U\right\|_{\infty})\}\subset\mathbb{R}^2$ of the radial solutions of the Emden-Fowler equation on $S_Θ$ $Δ_{\mathbb{S}^N}U+U^p=0$ in $S_Θ$, $U=0$ on $\partial S_Θ$, $U>0$ in $S_Θ$, where $p>1$. Among other things, we prove the following: For each $p>p_{\rm S}:=(N-2)/(N+2)$, there exists $\underlineΘ\in(0,π)$ such that the problem has a radial solution for $Θ\in(\underlineΘ,π)$ and has no radi…
We give a new approach on general systems of the form \[(G){[c]{c}% -Δ_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=ε_{1}|x| ^{a}u^{s}v^δ, -Δ_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla u)=ε_{2}|x|^{b}u^μv^{m},\] where $Q,p,q,δ,μ,s,m,$ $a,b$ are real parameters, $Q,p,q\neq1,$ and $ε_{1}=\pm1,$ $ε_{2}=\pm1.$ In the radial case we reduce the problem to a quadratic system of order 4, of Kolmogorov type. Then we obtain new local and global existence or nonexistence results. In the case $ε_{1}=ε_{2}=1,$ we also describe the behaviour of the ground states in two cases where the system is va…
We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $Ω$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial Ω$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $Ω$ is star-shaped around 0). However, in contrast to the non-satisfacto…
Radial solutions of quasilinear Hardy--Emden--Fowler equations: exact decay rates, Delaunay solutions, and nonexistence
Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam（经济数学学院，经济与法律大学）；Vietnam National University, Ho Chi Minh City, Vietnam（越南国立大学）
The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler equation and to the Van der Pol type equation.
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.
Let $N\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation $Δu + u^p = 0$ in $\mathbb{R}^N$ and recall the classical Liouville type theorem: if $u$ is a non-negative classical solution of the Lane-Emden equation, then $u \equiv 0$. The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of J. Serrin and H. Zou, originally used for the Lane-Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville type t…
The paper deals with the study of a Lane-Emden-Fowler equation with Dirichlet boundary condition and variable potential functions. The analysis developed in this paper combines monotonicity methods with variational arguments. Remark (April 21, 2020): Our results were later studied also by D.-P. Covei, Quasilinear problems with the competition between convex and concave nonlinearities and variable potentials, Internat. J. Math. 24:1 (2013), art. 1350005), arXiv:1104.4626v1.
Nonradial stable solutions near the Joseph--Lundgren threshold
University of Science and Technology of China（中国科学技术大学）；Beijing Technology and Business University（北京工商大学）；Chinese University of Hong Kong（香港中文大学）；University of Macau（澳门大学）
In this paper we study the Lane-Emden-Fowler equation $$(P)_ε \{Δu+|u|^{q-2}u=0 \ \hbox{in}\ \mathcal D_ε, u=0 \ \hbox{on}\ \partial\mathcal D_ε.$$ Here $\mathcal D_ε= \mathcal D \setminus \{x \in \mathcal D \ : \ \mathrm{dist}(x,Γ_\ell)\le ε\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $Γ_\ell$ is an $\ell-$dimensional closed manifold such that $Γ_\ell \subset \mathcal D$ with $1\le \ell \le N-3$ and $q={2(N-\ell)\over N-\ell-2}$. We prove that, under some symmetry assumptions, the number of sign changing solutions to $(P)_ε$ increases as $ε$ goes to zero.
We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation $$ -Δu+ u=u^{2^*-1}+λu^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2<q<2^*$ and $λ>0$ is a parameter. It is known that as $λ\to 0$, after a rescaling the ground state solutions of the equation converge to a particular solution of the critical Emden-Fowler equation $-Δu=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension $N=…
We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)Δu+ λu= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $λ\to 0$ and $λ\to +\infty$, where $N=3$ or $N= 4$, $2<q\le p\le 2^*$, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $a>0$, $b\ge 0$ are constants and $λ>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $λ\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-Δu+u=u^{q-1}$ a…
In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane-Emden-Fowler equations with singular nonlinearities and subquadratic convection term. We study the combined effects of various terms involved in these problems: sublinear or superlinear nonlinearities, singular nonlinear terms, convecti…
The objective of this paper is to obtain qualitative characteristics of multi-bubble solutions to the Lane-Emden-Fowler equations with slightly subcritical exponents given any dimension $n \ge 3$. By examining the linearized problem at each $m$-bubble solution, we provide a number of estimates on the first $(n+2)m$-eigenvalues and their corresponding eigenfunctions. Specifically, we present a new proof of the classical theorem due to Bahri-Li-Rey (Calc. Var. Partial Differential Equations 3 (1995) 67-93) which states that if $n \ge 4$, then the Morse index of a multi-bubble solution is governe…
In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic function, i.e., the solution of $Δ_{p}u=0,\ x\inΩ$, which extends the well-established results of the interior estimates of the gradient of harmonic function. We then get singularity and decay estimates of the sign changing solution of Lane-Emden-Fowler type p-Laplace equation $-Δ_{p}u=|u|^{λ-1}u, \ x\inΩ$, which are then generalized for the equation with gen…
We classify the local asymptotic behavior of positive singular solutions to a class of subcritical sixth order equations on the punctured ball. Initially, using a version of the integral moving spheres technique, we prove that solutions are asymptotically radially symmetric solutions with respect to the origin. We divide our approach into some cases concerning the growth of nonlinearity. In general, we use an Emden--Fowler change of variables to translate our problem to a cylinder. In the lower critical regime, this is not enough, thus, we need to introduce a new notion of change of variables.…
We study qualitative properties for nonnegative solutions to a conformally invariant coupled system of fourth order equations involving critical exponents. For solutions defined in the punctured space, there exist essentially two cases to analyze. If the origin is a removable singularity, we prove that non-singular solutions are rotationally invariant and weakly positive. More precisely, they are the product of a fourth order spherical solution by a unit vector with nonnegative coordinates. If the origin is a non-removable singularity, we show that the solutions are radially symmetric and stro…
We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an in- definite weight on the nonlinearity f (u, r). In particular we are interested in the case where f is positive in a ball and negative outside, or in the re- versed situation. We extend the approach to elliptic equations in presence of Hardy potentials. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.
Optimal Rigidity Results for the $k$-Hessian Equation of Lane--Emden Type
Beihang University（北京航空航天大学）；University of Macau（澳门大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity, solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients. On the technical level, we combine integral sliding methods and qualitative analysis of ODEs, based on a conservation of energy result, to perform a topological two-parameter shooting technique. We first use the integral representation …
We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-Δ)^s u+εu &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \] where $s\in(0,1)$, $N>2s$, $q>p>2$ are fixed parameters and $ε>0$ is a vanishing parameter. For $ε>0$ small, we prove the existence of a ground state solution and show that any positive solution of above problem is a classical solution and radially symmetric and symmetric decreasing. We also obtain the decay rate of solution at infinity. Next, we study the asymptotic be…
We are concerned with the half-space Dirichlet problem \[\left\{\begin{array}{ll} -Δv+v=|v|^{p-1}v & \textrm{in}\ \mathbb{R}^N_+, v=c\ \textrm{on}\ \partial\mathbb{R}^N_+, &\lim_{x_N\to \infty}v(x',x_N)=0\ \textrm{uniformly in}\ x'\in\mathbb{R}^{N-1}, \end{array}\right. \] where $\mathbb{R}^N_+=\{x\in \mathbb{R}^N \ : \ x_N>0\}$ for some $N\geq 2$, and $p>1$, $c>0$ are constants. It was shown recently by Fernandez and Weth [Math. Ann. (2021)] that there exists an explicit number $c_p\in (1,\sqrt{e})$, depending only on $p$, such that for $0<c<c_p$ there are infinitely many bounded positive sol…
Consider the following coupled elliptic system of equations \begin{equation*} \label{} (-Δ)^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} \ \ \mathbb{R}^n , \end{equation*} where $0<s\le 2$, $p>1$, $m\ge1$, $u=(u_i)_{i=1}^m$ and $u_i:\mathbb R^n\to \mathbb R$. The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when $m=1$ and $s=1$, Gidas and Spruck in \cite{gs} and later Caffarelli, Gidas …
