Local and global behavior of solutions to 2 D-elliptic equation with exponentially-dominated nonlinearities

2021 ◽  
pp. 1-30
Author(s):  
Takashi Suzuki
Keyword(s):  
Growth Rate ◽  
Pattern Formation ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Uniform Estimate ◽  
Global Behavior ◽  
Semilinear Elliptic ◽  
Residual Part ◽  
The Family ◽  
Two Space Dimensions

We study the family of blowup solutions to semilinear elliptic equations in two-space dimensions with exponentially-dominated nonnegative nonlinearities. Such a family admits an exclusion of the boundary blowup, finiteness of blowup points, and pattern formation. Then, Hamiltonian control of the location of blowup points, residual vanishing, and mass quantization arise under the estimate from below of the nonlinearity. Finally, if the principal growth rate of nonlinearity is exactly exponential and the residual part has a gap relative to this term, there is a locally uniform estimate of the solution which ensures its asymptotic non-degeneracy.

scholarly journals Microscopic asymptotics for solutions of some semilinear Elliptic equations

1995 ◽  
Vol 138 ◽  
pp. 33-50
Author(s):  
Takayoshi Ogawa ◽  
Takashi Suzuki
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Classical Solutions ◽  
Semilinear Elliptic ◽  
The Family

In our previous work [8], we picked up the elliptic equation(1) with the nonlinearity f(u) ⊇ 0 in C1. We studied the asymptotics of the family {(λ, u(x))} of classical solutions satisfying(2)

Positive Solutions for Elliptic Equations With Supercritical Nonlinearity

2009 ◽  
Vol 9 (3) ◽  
Author(s):  
Paulo Rabelo
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Auxiliary Problem ◽  
Semilinear Elliptic Equation ◽  
Minimax Methods ◽  
Semilinear Elliptic ◽  
Priori Estimates ◽  
Supercritical Nonlinearity

AbstractIn this paper minimax methods are employed to establish the existence of a bounded positive solution for semilinear elliptic equation of the form−∆u + V (x)u = P(x)|u|where the nonlinearity has supercritical growth and the potential can change sign. The solutions of the problem above are obtained by proving a priori estimates for solutions of a suitable auxiliary problem.

scholarly journals Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

2018 ◽  
Vol 8 (1) ◽  
pp. 995-1003 ◽  
Author(s):  
Marius Ghergu ◽  
Sunghan Kim ◽  
Henrik Shahgholian
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Nonnegative Solution ◽  
Removable Singularity ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Isolated Singularity ◽  
Semilinear Elliptic

Abstract We study the semilinear elliptic equation -\Delta u=u^{\alpha}\lvert\log u|^{\beta}\quad\text{in }B_{1}\setminus\{0\}, where {B_{1}\subset{\mathbb{R}}^{n}} , with {n\geq 3} , {\frac{n}{n-2}<\alpha<\frac{n+2}{n-2}} and {-\infty<\beta<\infty} . Our main result establishes that the nonnegative solution {u\in C^{2}(B_{1}\setminus\{0\})} of the above equation either has a removable singularity at the origin or it behaves like u(x)=A(1+o(1))|x|^{-\frac{2}{\alpha-1}}\Bigl{(}\log\frac{1}{|x|}\Big{)}^{-% \frac{\beta}{\alpha-1}}\quad\text{as }x\rightarrow 0, with {A=[(\frac{2}{\alpha-1})^{1-\beta}(n-2-\frac{2}{\alpha-1})]^{\frac{1}{\alpha-1% }}.}

scholarly journals Super and subsolutions for elliptic equations on all ofℝn

2002 ◽  
Vol 32 (1) ◽  
pp. 41-46
Author(s):  
G. A. Afrouzi ◽  
H. Ghasemzadeh
Keyword(s):  
Population Genetics ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Asymptotic Properties ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic

By construction sub and supersolutions for the following semilinear elliptic equation−△u(x)=λg(x)f(u(x)),x∈ℝnwhich arises in population genetics, we derive some results about the theory of existence of solutions as well as asymptotic properties of the solutions for everynand for the functiong:ℝn→ℝsuch thatgis smooth and is negative at infinity.

A Singular Semilinear Elliptic Equation with a Variable Exponent

2016 ◽  
Vol 16 (3) ◽  
Author(s):  
José Carmona ◽  
Pedro J. Martínez-Aparicio
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Model Problem ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Bounded Set ◽  
Semilinear Elliptic

AbstractIn this paper we consider singular semilinear elliptic equations with a variable exponent whose model problem isHere Ω is an open bounded set of

scholarly journals On the exact multiplicity of stable ground states of non-Lipschitz semilinear elliptic equations for some classes of starshaped sets

2019 ◽  
Vol 9 (1) ◽  
pp. 1046-1065 ◽  
Author(s):  
J.I. Díaz ◽  
J. Hernández ◽  
Y.Sh. Ilyasov
Keyword(s):  
Elliptic Equation ◽  
Compact Support ◽  
Elliptic Equations ◽  
Variational Approach ◽  
Semilinear Elliptic Equation ◽  
Number Of Solutions ◽  
Semilinear Elliptic ◽  
Starshaped Sets ◽  
Stable Solutions ◽  
Exact Multiplicity

Abstract We prove the exact multiplicity of flat and compact support stable solutions of an autonomous non-Lipschitz semilinear elliptic equation of eigenvalue type according to the dimension N and the two exponents, 0 < α < β < 1, of the involved nonlinearites. Suitable assumptions are made on the spatial domain Ω where the problem is formulated in order to avoid a possible continuum of those solutions and, on the contrary, to ensure the exact number of solutions according to the nature of the domain Ω. Our results also clarify some previous works in the literature. The main techniques of proof are a Pohozhaev’s type identity and some fibering type arguments in the variational approach.

scholarly journals A uniqueness result for some singular semilinear elliptic equations

2016 ◽  
Vol 18 (06) ◽  
pp. 1550084 ◽  
Author(s):  
Annamaria Canino ◽  
Berardino Sciunzi
Keyword(s):  
Elliptic Equation ◽  
Boundary Conditions ◽  
Open Subset ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Uniqueness Result ◽  
Dirichlet Boundary Conditions ◽  
Semilinear Elliptic ◽  
Bounded Open Subset

Given [Formula: see text] a bounded open subset of [Formula: see text], we consider non-negative solutions to the singular semilinear elliptic equation [Formula: see text] in [Formula: see text], under zero Dirichlet boundary conditions. For [Formula: see text] and [Formula: see text], we prove that the solution is unique.

Existence of solutions to a class of semilinear elliptic equations involving general subcritical growth

2014 ◽  
Vol 144 (4) ◽  
pp. 809-818 ◽  
Author(s):  
Yong-Yi Lan ◽  
Chun-Lei Tang
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Semilinear Elliptic Equation ◽  
Existence Results ◽  
Subcritical Growth ◽  
Analysis Techniques ◽  
Semilinear Elliptic

In this paper, we consider the semilinear elliptic equation −Δu = λf(x,u) with the Dirichlet boundary value, and under suitable assumptions on the nonlinear term f with a more general growth condition. Some existence results of solutions are given for all λ > 0 via the variational method and some analysis techniques.

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