Infinitely many non-radial solutions for the polyharmonic Hénon equation with a critical exponent

2017 ◽  
Vol 147 (2) ◽  
pp. 371-396 ◽  
Author(s):  
Yuxia Guo ◽  
Bo Li ◽  
Yi Li
Keyword(s):  
Critical Exponent ◽  
Unit Ball ◽  
Positive Solutions ◽  
Bounded Function ◽  
Radial Solutions ◽  
Hénon Equation ◽  
Image Position

We study the following polyharmonic Hénon equation:where (m)* = 2N/(N – 2m) is the critical exponent, B1(0) is the unit ball in ℝN, N ⩾ 2m + 2 and K(|y|) is a bounded function. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

scholarly journals A priori bounds and multiplicity of positive solutions for p-Laplacian Neumann problems with sub-critical growth

2019 ◽  
Vol 150 (1) ◽  
pp. 73-102 ◽  
Author(s):  
Alberto Boscaggin ◽  
Francesca Colasuonno ◽  
Benedetta Noris
Keyword(s):  
Positive Solutions ◽  
Critical Power ◽  
A Priori ◽  
Oscillatory Behaviour ◽  
Radial Solutions ◽  
A Priori Bounds ◽  
Shooting Technique ◽  
Neumann Problems ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

AbstractLet 1 < p < +∞ and let Ω ⊂ ℝN be either a ball or an annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris, ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann problems of the type $-\Delta _pu = f(u),\quad u > 0\,{\rm in }\,\Omega ,\quad \partial _\nu u = 0\,{\rm on }\,\partial \Omega .$We suppose that f(0) = f(1) = 0 and that f is negative between the two zeros and positive after. In case Ω is a ball, we also require that f grows less than the Sobolev-critical power at infinity. We prove a priori bounds of radial solutions, focussing in particular on solutions which start above 1. As an application, we use the shooting technique to get existence, multiplicity and oscillatory behaviour (around 1) of non-constant radial solutions.

Existence of multiple positive solutions for a semilinear equation with critical exponent

1992 ◽  
Vol 122 (1-2) ◽  
pp. 161-175 ◽  
Author(s):  
Deng Yinbing
Keyword(s):  
Critical Exponent ◽  
Unique Solution ◽  
Positive Solutions ◽  
Positive Constant ◽  
Multiple Positive Solutions ◽  
Semilinear Equation ◽  
Image Position

SynopsisIn this paper we discuss the problemWe show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).

Estimate, existence and nonexistence of positive solutions of Hardy–Hénon equations

2021 ◽  
pp. 1-24
Author(s):  
Xiyou Cheng ◽  
Lei Wei ◽  
Yimin Zhang
Keyword(s):  
Boundary Condition ◽  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Dirichlet Boundary Condition ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Existence And Nonexistence ◽  
Hénon Equation ◽  
Image Position

We consider the boundary Hardy–Hénon equation \[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \] where $B_1(0)\subset \mathbb {R}^{N}$   $(N\geq 3)$ is a ball of radial $1$ centred at $0$ , $p>0$ and $\alpha \in \mathbb {R}$ . We are concerned with the estimate, existence and nonexistence of positive solutions of the equation, in particular, the equation with Dirichlet boundary condition. For the case $0< p<({N+2})/({N-2})$ , we establish the estimate of positive solutions. When $\alpha \leq -2$ and $p>1$ , we give some conclusions with respect to nonexistence. When $\alpha >-2$ and $1< p<({N+2})/({N-2})$ , we obtain the existence of positive solution for the corresponding Dirichlet problem. When $0< p\leq 1$ and $\alpha \leq -2$ , we show the nonexistence of positive solutions. When $0< p<1$ , $\alpha >-2$ , we give some results with respect to existence and uniqueness of positive solutions.

scholarly journals THE ROBIN PROBLEM FOR THE HÉNON EQUATION

2013 ◽  
Vol 88 (1) ◽  
pp. 1-11
Author(s):  
HAIYANG HE
Keyword(s):  
Symmetry Breaking ◽  
Unit Ball ◽  
Ground State ◽  
Existence Results ◽  
Robin Problem ◽  
Ground State Solutions ◽  
Hénon Equation ◽  
Image Position

AbstractIn this paper, we consider the following Robin problem:$$\begin{eqnarray*}\displaystyle \left\{ \begin{array}{ @{}ll@{}} \displaystyle - \Delta u= \mid x{\mathop{\mid }\nolimits }^{\alpha } {u}^{p} , \quad & \displaystyle x\in \Omega , \\ \displaystyle u\gt 0, \quad & \displaystyle x\in \Omega , \\ \displaystyle \displaystyle \frac{\partial u}{\partial \nu } + \beta u= 0, \quad & \displaystyle x\in \partial \Omega , \end{array} \right.&&\displaystyle\end{eqnarray*}$$where$\Omega $is the unit ball in${ \mathbb{R} }^{N} $centred at the origin, with$N\geq 3$,$p\gt 1$,$\alpha \gt 0$,$\beta \gt 0$, and$\nu $is the unit outward vector normal to$\partial \Omega $. We prove that the above problem has no solution when$\beta $is small enough. We also obtain existence results and we analyse the symmetry breaking of the ground state solutions.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

On the blow-up of solutions of a convective reaction diffusion equation

1993 ◽  
Vol 123 (3) ◽  
pp. 433-460 ◽  
Author(s):  
J. Aguirre ◽  
M. Escobedo
Keyword(s):  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Scalar Function ◽  
Reaction Diffusion Equation ◽  
Semilinear Parabolic Equation ◽  
The Cauchy Problem ◽  
Image Position

SynopsisWe study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equationwhere u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.

scholarly journals POSITIVE SOLUTIONS OF A SECOND-ORDER NEUMANN BOUNDARY VALUE PROBLEM WITH A PARAMETER

2012 ◽  
Vol 86 (2) ◽  
pp. 244-253 ◽  
Author(s):  
YANG-WEN ZHANG ◽  
HONG-XU LI
Keyword(s):  
Boundary Value Problem ◽  
Positive Solutions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Existence And Uniqueness Theorem ◽  
Neumann Boundary Value Problem ◽  
Image Position ◽  
Continuous Dependence Of Solutions

AbstractIn this paper, we consider the Neumann boundary value problem with a parameter λ∈(0,∞): By using fixed point theorems in a cone, we obtain some existence, multiplicity and nonexistence results for positive solutions in terms of different values of λ. We also prove an existence and uniqueness theorem and show the continuous dependence of solutions on the parameter λ.

Kolmogorov, Linear and Pseudo-Dimensional Widths of Classes of s-Monotone Functions in 𝕃p, 0 < p < 1

2008 ◽  
Vol 51 (2) ◽  
pp. 236-248
Author(s):  
Victor N. Konovalov ◽  
Kirill A. Kopotun
Keyword(s):  
Unit Ball ◽  
Finite Interval ◽  
Divided Differences ◽  
Monotone Functions ◽  
Image Position

AbstractLet Bp be the unit ball in 𝕃p, 0 < p < 1, and let , s ∈ ℕ, be the set of all s-monotone functions on a finite interval I, i.e., consists of all functions x : I ⟼ ℝ such that the divided differences [x; t0, … , ts] of order s are nonnegative for all choices of (s + 1) distinct points t0, … , ts ∈ I. For the classes Bp := ∩ Bp, we obtain exact orders of Kolmogorov, linear and pseudo-dimensional widths in the spaces , 0 < q < p < 1:

scholarly journals Radial Symmetry and Monotonicity of Solutions to a System Involving Fractional p-Laplacian in a Ball

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Linfen Cao ◽  
Xiaoshan Wang ◽  
Zhaohui Dai
Keyword(s):  
Nonlinear System ◽  
Unit Ball ◽  
Positive Solutions ◽  
Direct Method ◽  
Radial Symmetry ◽  
Continuous Functions ◽  
Method Of Moving Planes ◽  
Moving Planes

In this paper, we study a nonlinear system involving the fractional p-Laplacian in a unit ball and establish the radial symmetry and monotonicity of its positive solutions. By using the direct method of moving planes, we prove the following result. For 0<s,t<1,p>0, if u and v satisfy the following nonlinear system -Δpsux=fvx;  -Δptvx=gux,  x∈B10;  ux,vx=0,  x∉B10. and f,g are nonnegative continuous functions satisfying the following: (i) f(r) and g(r) are increasing for r>0; (ii) f′(r)/rp-2, g′(r)/rp-2 are bounded near r=0. Then the positive solutions (u,v) must be radially symmetric and monotone decreasing about the origin.

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