scholarly journals On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers

2008 ◽  
Vol 2008 ◽  
pp. 1-6 ◽  
Author(s):  
Claudianor O. Alves ◽  
Marco A. S. Souto
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equation ◽  
Existence Of Solution ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic

We prove that the semilinear elliptic equation−Δu=f(u), inΩ,u=0, on∂Ωhas a positive solution when the nonlinearityfbelongs to a class which satisfiesμtq≤f(t)≤Ctpat infinity and behaves liketqnear the origin, where1<q<(N+2)/(N−2)ifN≥3and1<q<+∞ifN=1,2. In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth ofp.

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% }}.}

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 Multiple Positive Solutions for Semilinear Elliptic Equations in Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Weight Functions ◽  
Multiple Positive Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Multiplicity Of Positive Solutions

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation in , , where , if , if ), , satisfy suitable conditions, and may change sign in .

Existence of 2–Nodal Solutions for Semilinear Elliptic Equations in Unbounded Domains

2010 ◽  
Vol 10 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Tiexiang Li ◽  
Huei-li Lin ◽  
Tsung-fang Wu
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Unbounded Domains ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Nodal Solutions ◽  
Domain Shape ◽  
Semilinear Elliptic

AbstractIn this paper, we study the effect of domain shape on the existence of 2-nodal solutions for a semilinear elliptic equation involving non-odd nonlinearities.

Gradient estimates for semilinear elliptic equations

1985 ◽  
Vol 100 (1-2) ◽  
pp. 11-17
Author(s):  
Gary M. Lieberman
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Elliptic Equations ◽  
Boundary Data ◽  
Smooth Boundary ◽  
Semilinear Elliptic Equation ◽  
Semilinear Elliptic Equations ◽  
Gradient Estimates ◽  
Quadratic Growth ◽  
Semilinear Elliptic

SynopsisEstimates on the gradient of solutions to the Dirichlet problem for a semilinear elliptic equation are given when the nonlinearity in the equation is quadratic with respect to the gradient of the solution. These estimates extend results of F. Tomi to less smooth boundary data and results of the author to the full quadratic growth.

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 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.

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone

1992 ◽  
Vol 122 (1-2) ◽  
pp. 137-160
Author(s):  
Chie-Ping Chu ◽  
Hwai-Chiuan Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Semilinear Elliptic Equations ◽  
Spherically Symmetric ◽  
Semilinear Elliptic ◽  
Symmetry Properties

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

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