scholarly journals Singular solutions of a Hénon equation involving a nonlinear gradient term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Craig Cowan ◽  
Abdolrahman Razani
Keyword(s):  
Unit Ball ◽  
Explicit Solution ◽  
Singular Solutions ◽  
Gradient Term ◽  
Hénon Equation ◽  
Smooth Perturbation

<p style='text-indent:20px;'>Here, we consider positive singular solutions of</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{lcc} -\Delta u = |x|^\alpha |\nabla u|^p &amp; \text{in}&amp; \Omega \backslash\{0\},\\ u = 0&amp;\text{on}&amp; \partial \Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a small smooth perturbation of the unit ball in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> are parameters in a certain range. Using an explicit solution on <inline-formula><tex-math id="M5">\begin{document}$ B_1 $\end{document}</tex-math></inline-formula> and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.</p>

scholarly journals A semilinear problem with a gradient term in the nonlinearity

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ignacio Guerra
Keyword(s):  
Unit Ball ◽  
Radial Solution ◽  
The Other ◽  
Smooth Domain ◽  
Gradient Term ◽  
Radial Solutions ◽  
Bounded Smooth Domain ◽  
Other Hand ◽  
Bounded Smooth ◽  
Semilinear Problem

<p style='text-indent:20px;'>We consider the following semilinear problem with a gradient term in the nonlinearity</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} -\Delta u = \lambda \frac{(1+|\nabla u|^q)}{(1-u)^p}\quad\text{in}\quad\Omega,\quad u&gt;0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda,p,q&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> be a bounded, smooth domain in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>. We prove that when <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a unit ball and <inline-formula><tex-math id="M5">\begin{document}$ p = 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ q\in (0,q^*(N)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ q^*(N)\in (1,2) $\end{document}</tex-math></inline-formula>, we have infinitely many radial solutions for <inline-formula><tex-math id="M8">\begin{document}$ 2\leq N&lt;2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda = \tilde \lambda $\end{document}</tex-math></inline-formula>. On the other hand, for <inline-formula><tex-math id="M10">\begin{document}$ N&gt;2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> there exists a unique radial solution for <inline-formula><tex-math id="M11">\begin{document}$ 0&lt;\lambda&lt;\tilde \lambda $\end{document}</tex-math></inline-formula>.</p>

Least Energy Solutions and Group Invariant Solutions of the Hénon Equation

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Ryuji Kajikiya
Keyword(s):  
Unit Ball ◽  
Radial Symmetry ◽  
Coefficient Function ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Least Energy Solution ◽  
Hénon Equation ◽  
Group Invariant Solutions ◽  
Least Energy Solutions ◽  
Least Energy

AbstractIn this paper we study the generalized Hénon equation in the unit ball, where the coefficient function may or may not change its sign. We prove that the least energy solution is not radial and moreover we show the existence of a group invariant positive solution without radial symmetry.

scholarly journals PERTURBING SINGULAR SOLUTIONS OF THE GELFAND PROBLEM

2007 ◽  
Vol 09 (05) ◽  
pp. 639-680 ◽  
Author(s):  
J. DÁVILA ◽  
L. DUPAIGNE
Keyword(s):  
Unit Ball ◽  
Singular Solution ◽  
Dirichlet Condition ◽  
Extremal Solution ◽  
Singular Solutions ◽  
The One ◽  
Gelfand Problem ◽  
Small Deformations

The equation -Δu = λeu posed in the unit ball B ⊆ ℝN, with homogeneous Dirichlet condition u|∂B = 0, has the singular solution [Formula: see text] when λ = 2(N - 2). If N ≥ 4 we show that under small deformations of the ball there is a singular solution (u,λ) close to (U,2(N - 2)). In dimension N ≥ 11 it corresponds to the extremal solution — the one associated to the largest λ for which existence holds. In contrast, we prove that if the deformation is sufficiently large then even when N ≥ 10, the extremal solution remains bounded in many cases.

scholarly journals Nonradial Solutions for the Hénon Equation Close to the Threshold

2019 ◽  
Vol 19 (4) ◽  
pp. 757-770 ◽  
Author(s):  
Pablo Figueroa ◽  
Sérgio L. N. Neves
Keyword(s):  
Positive Integer ◽  
Unit Ball ◽  
Hénon Equation

AbstractWe consider the Hénon problem\left\{\begin{aligned} &\displaystyle{-}\Delta u=\lvert x\rvert^{\alpha}u^{% \frac{N+2+2\alpha}{N-2}-\varepsilon}&&\displaystyle\phantom{}\text{in }B_{1},% \\ &\displaystyle u>0&&\displaystyle\phantom{}\text{in }B_{1},\\ &\displaystyle u=0&&\displaystyle\phantom{}\text{on }\partial B_{1},\end{% aligned}\right.where {B_{1}} is the unit ball in {\mathbb{R}^{N}} and {N\geqslant 3}. For {\varepsilon>0} small enough, we use α as a parameter and prove the existence of a branch of nonradial solutions that bifurcates from the radial one when α is close to an even positive integer.

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.

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.

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