Structure Of Positive Radial Solutions Research Articles

Structure of positive radial solutions of a quasilinear elliptic problem with singular nonlinearity

ABSTRACTWe study the structure of radial solutions of the quasilinear elliptic problem (Q) where , , and . It is seen that (Q) admits a global continuum of radial solutions . Moreover, there is such that has infinitely many turning points for and does not admit any turning point for .

Positive solutions of semilinear elliptic equations: a dynamical approach

This paper is devoted to the study of the structure of positive radial solutions for the following semi-linear equation: $$\Delta u + f(u,|x|)=0 .$$ We require $f$ to be nonnegative and to exhibit both subcritical and supercritical behavior with respect to the Sobolev critical exponent. More precisely we assume that $f$ is subcritical for $u$ small and $|x|$ large and supercritical for $u$ large and $|x|$ small, and we give existence and non-existence results for ground states regular and singular, with either fast or slow decay. We find a surprisingly rich structure, which is characterized by two different patterns of bifurcations. We perform a Fowler transformation and we use a dynamical approach, exploiting some ideas borrowed from Bamon, Del Pino, and Flores, combining them with the use of the translation of the Pohozaev function for this dynamical context.

On the elliptic equation Δu+K <I>u</I><SUP><I>p</I></SUP> = 0 in <I>$\mathbb{R}$</I><SUP><I>n</I></SUP>

This paper deals with the elliptic equation Δu+K(|x|) up = 0 in $\mathbb{R}$n\{0} when $r^{-l}K(r)$ for $l>-2$ behaves monotonically near $r=0$ or $\infty$ with $r=|x|$. By the method of phase plane, we present a new proof for the structure of positive radial solutions, and analyze the asymptotic behavior at $\infty$. We also employ the approach to classify singular solutions in terms of the asymptotic behavior at $0$. In particular, when $p=\frac{n+2+2l}{n-2}$, we establish the uniqueness of solutions with asymptotic self-similarity at $0$ and at $\infty$, and the existence of multiple solutions of Delaunay-Fowler type at $0$ and $\infty$.

Positive Solutions for Semilinear Elliptic Equations: Two Simple Models with Several Bifurcations

In this paper we analyze the structure of positive radial solutions for the following semi-linear equations: $$ \Delta u + f(u,|{\bf x}|)=0 $$ where \({{\bf x}\in \mathbb{R}^n}\) and f is superlinear. In fact we just consider two very special non-linearities, i.e. $$\label{uno} f(u,|{\bf x}|) = u|u|^{q-2}\max\{|{\bf x}|^{\delta^s}, |{\bf x}|^{\delta^u}\}\; \quad -2 < \delta^u < \lambda^{\ast} < \delta^s < \lambda_{\ast}, \quad \quad \quad (0.1) $$ i.e. f is supercritical for |x| small and subcritical for |x| large, and $$\label{due} f(u)= \max\{u|u|^{q^s-2}, u|u|^{q^u-2}\}, \quad 2_{\ast} < q^s <2 ^{\ast} < q^u \quad \quad \quad \quad (0.2) $$ i.e. f is subcritical for u small and supercritical for u large. We find a surprisingly rich structure for both the non-linearities, similar to the one detected by Bamon et al. for \({f=u^{q^u-1}+u^{q^s-1}}\) when 2* < qs < 2* < qu. More precisely if we fix qs and we let qu vary in (0.2) we find that there are no ground states for qu large, and an arbitrarily large number of ground states with fast decay as qu approaches 2*. We also find the symmetric result when we fix qu and let qs vary. We also prove the existence of a further resonance phenomenon which generates small windows with a large number of ground states with fast decay. Similar results hold for (0.1).

On the structure of positive solutions to an elliptic problem arising in thin film equations

We consider the structure of positive radial solutions of Δ u + u − α − u − β = 0 in B R , u = 0 on ∂ B R , where B R is a ball in R N with radius R. When 0 < α < β < 1 , we show that there exists R ∗ > 0 such that when R > R ∗ , the Dirichlet problem has exactly two radial solutions; when R = R ∗ , the solution is unique and there is no solution for R < R ∗ . When 0 < β < α < 1 , we show that for any R > 0 , the radial solution exists and is unique.

On the structure of positive solutions to an elliptic problem with a singular nonlinearity

We consider the following boundary value problem Δ u = λ [ u − p − u − q ] in Ω , u = κ ∈ ( 0 , ∞ ) on ∂ Ω , where p > q > 0 and Ω is a bounded smooth domain in R N ( N ⩾ 2 ). It is shown that, for κ > 1 , there is at least one solution u λ of this problem satisfying u λ > κ in Ω for any λ > 0 . The profiles of u λ for λ → 0 and λ → ∞ are analyzed. For κ = 1 , it is clear that u λ ≡ 1 is a constant solution to the problem. We also present the branch of solutions satisfying 0 < u λ < 1 in Ω for the equation with κ = 1 . We obtain more information on the structure of positive radial solutions of the problem when Ω is an annulus.

A note on the positive solutions of an inhomogeneous elliptic equation on [formula omitted

This paper is contributed to the elliptic equation (0.1) Δ u + K ( | x | ) u p + μ f ( | x | ) = 0 , where p > 1 , x ∈ R n , n ⩾ 3 , Δ = ∑ i = 1 n ∂ 2 ∂ x i 2 and μ ⩾ 0 is a constant. We study the structure of positive radial solutions of (0.1) and obtain the uniqueness of solution decaying faster than r − m at ∞ if μ is small enough under some assumptions on K and f, where m is the slow decay rate.

Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems

We apply the “monotone separation of graphs” technique of L.A. Peletier and J. Serrin [L.A. Peletier, J. Serrin, Uniqueness of positive solutions of semilinear equations in R n , Arch. Ration. Mech. Anal. 81 (2) (1983) 181–197; L.A. Peletier, J. Serrin, Uniqueness of nonnegative solutions of semilinear equations in R n , J. Differential Equations 61 (3) (1986) 380–397], as developed further by L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202], to the question of exact multiplicity of positive solutions for a class of semilinear equations on a unit ball in R n . We also observe that using P. Pucci and J. Serrin [P. Pucci, J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (2) (1998) 501–528] improvement of a certain identity of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202] produces a short proof of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179–202] result on the uniqueness of positive solution of ( 1 < p , q < n + 2 n − 2 ) Δ u + u p + u q = 0 for | x | < 1 , u = 0 when | x | = 1 .

Existence of global solutions for a semilinear parabolic Cauchy problem

In this paper, we consider the parabolic equation $w_t=\Delta w+|x|^{l} w^p$, $x \in {\bf R}^n$, $t>0$ with $w(x, 0)=f(x)$ and show the existence of global solution if $1+(2+l)/n < p <(n+2+2l)/(n-2)$ for each $n \geq 3$ and $l \in (-2, l^*]$, where $l^*=0$ if $n \geq 4$ and $l^*=\sqrt3-1$ if $n=3$. In order to prove this result, we need an upper solution for this Cauchy problem. If $f(x)$ satisfies some condition, then we can show the existence of upper solution by investigating the structure of positive radial solutions for related elliptic equation which has a gradient term.

On the Cauchy problem for a reaction–diffusion equation with a singular nonlinearity

We consider the following Cauchy problem with a singular nonlinearity ( P ) u t = Δ u − u − ν , x ∈ R n , t > 0 , ν > 0 , u | t = 0 = ϕ ∈ C LB ( R n ) with n ⩾ 3 (and ϕ having a positive lower bound). We find some conditions on the initial value ϕ such that the local solutions of ( P ) vanish in finite time. Meanwhile, we obtain optimal conditions on ϕ for global existence and study the large time behavior of those global solutions. In particular, we prove that if ν > 0 and n ⩾ 3 , ϕ ( x ) ⩾ γ u s ( x ) = γ [ 2 ν + 1 ( n − 2 + 2 ν + 1 ) ] − 1 / ( ν + 1 ) | x | 2 / ( ν + 1 ) , where u s is a singular equilibrium of ( P ) and γ > 1 , then ( P ) has a (unique) global classical solution u with u ⩾ γ u s and u ( x , t ) ⩾ ( ν + 1 ) 1 / ( ν + 1 ) ( γ ν + 1 − 1 ) 1 / ( ν + 1 ) t 1 / ( ν + 1 ) . On the other hand, the structure of positive radial solutions of the steady-state of ( P ) is studied and some interesting properties of the positive solutions are obtained. Moreover, the stability and weakly asymptotic stability of the positive radial solutions of the steady-state of ( P ) are also discussed.

On the structure of positive radial solutions to an equation containing a p-Laplacian with weight

Let A , B : ( 0 , ∞ ) ↦ ( 0 , ∞ ) be two given weight functions and consider the equation ( P ) - div A ( | x | ) | ∇ u | p - 2 ∇ u = B ( | x | ) | u | q - 2 u , x ∈ R n , where q > p > 1 . By considering positive radial solutions to this equation that are bounded, we are led to study the initial value problem - a ( r ) | u ′ | p - 2 u ′ ′ = b ( r ) ( u + ) q - 1 , r ∈ ( 0 , ∞ ) , u ( 0 ) = α > 0 , lim r → 0 a ( r ) | u ′ ( r ) | p - 1 = 0 , where a ( r ) = r ( N - 1 ) A ( r ) and b ( r ) = r ( N - 1 ) B ( r ) . By means of two key functions m and B q defined below, we obtain several new results that allow us to classify solutions to this initial value problem as being respectively crossing, slowly decaying, or rapidly decaying. We also generalize several results in Clément et al. (Asymptotic Anal. 17 (1998) 13–29), Kawano et al. (Funkcial. Ekvac 36 (1993) 121–145), Yanagida and Yotsutani (Arch. Rational Mech. Anal. 124 (1993) 239–259), Yanagida and Yotsutani (J. Differential Equations 115 (1995) 477–502), Yanagida and Yotsutani (Arch. Rational Mech. Anal. 134 (1996) 199–226).

Structure of positive radial solutions including singular solutions to Matukuma's equation

We consider positive solutions of Matukuma's equation, which is described by a nonlinear elliptic equation with a weight. Any radially symmetric solution of this equation is said to be regular or singular according to its behavior near the origin and infinity. We investigate the structure of positive radial regular and singular solutions.

Separation structure of positive radial solutions of a semilinear elliptic equation in [formula omitted

This paper deals with the following inter-connected subjects: (i) Separation of positive radial solutions is studied for the elliptic equation Δu+K(|x|)up=0 in Rn. In case r−ℓK(r) is decreasing, with suitable decay rate, to a positive constant as r→∞ for some ℓ>−2, the asymptotic behavior near ∞ of radial solutions is described in detail when n and p are large enough. When the equation has separation structure (any two positive radial solutions do not intersect), the obtained asymptotic behaviors decide natural topology for stability of positive radial solutions regarded as steady states of the corresponding semilinear heat equation. (ii) By using local separation of regular solutions, we establish the existence of a singular solution when every solution with positive initial data at 0 exists globally. (iii) Through Kelvin's transform, separation structure of regular solutions reflects that of singular solutions. We present a new exponent, which is critical in the study of separation and intersection of singular solutions.

On the structure of positive radial solutions for quasilinear equations in annular domains

We study the existence, multiplicity and nonexistence of positive radial solutions to boundary value problems for the quasilinear equation $\text{ div} \left ( A(| \nabla u|)\nabla u \right ) + \lambda h(|x|)f(u) =0$ in annular domains under general assumptions on the function $A(u)$. Various possible behaviors of the quotient $\frac{f(u)}{A(u)u}$ at zero and infinity are considered. We shall use fixed point theorems for operators on a Banach space.

Structure of Positive Radial Solutions to Scalar Field Equations with Harmonic Potential

We study a nonlinear elliptic equation with a harmonic potential−Δ u+(λ+|x|2)u−|u|p−1u=0,x∈Rn, (E) which is related to standing waves for a nonlinear Schrödinger equation. The existence of positive solution for (E) in some energy space can be shown by the standard variational method under n⩾1, λ>−n and 1<p<(n+2)/(n−2)+. However, the uniqueness is a delicate problem. In this paper, noting that all positive solutions of (E) in the energy space are radially symmetric about the origin, we study the structure of positive radial solutions to (E) and show the uniqueness of positive solution with finite energy when n⩾3.

Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems

We propose a method to investigate the structure of positive radial solutions to semilinear elliptic problems with various boundary conditions. It is already shown that the boundary value problems can be reduced to a canonical form by a suitable change of variables. We show structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.

UNIQUENESS AND GLOBAL STRUCTURE OF POSITIVE RADIAL SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS

Using only elementary analysis based on an important identity of Ni, Pucci and Serrin, and a maximum principle of Peletier and Serrin, we completely clarify the structure of positive radial solutions of m-Laplace equations. For a large class of nonlinearities, including polynomial, rational, exponential, logarithmic and trigonometric functions, we can then resolve problems of existence and uniqueness of ground states or positive radial solutions of the associated Dirichlet problem in a ball in R n .

Existence and Uniqueness of Fast Decay Entire Solutions of Quasilinear Elliptic Equations

We prove the existence and uniqueness of fast decay solutions and clarify the structure of positive radial solutions of the quasilinear elliptic equation Δmu+f(u)=0 in Rn, where Δm denotes the m-Laplace operator, and f has a supercritical growth for small u>0 and a subcritical growth for large u. Our proofs use only elementary arguments based on several variational identities and a maximum principle of Peletier and Serrin.

Structure of positive radial solutions to the Haraux-Weissler equation

Structure of positive radial solutions to the Haraux-Weissler equation

Uniqueness of positive radial solutions of $\Delta u+K(\vert x\vert )\gamma(u)=0$

We investigate the global structure of positive radial solutions of a semilinear elliptic equation $\Delta u+K(|x|)\gamma (u)=0$, and study the uniqueness of ground state solutions of this equation. Our discussion is based on a Pohozaev-type identity and some detailed investigation for the oscillatory and asymptotic behavior of the solutions and their variational functions.