Displaying similar documents to “Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition”

Multiplicity results for a class of concave-convex elliptic systems involving sign-changing weight functions

Honghui Yin, Zuodong Yang (2011)

Annales Polonici Mathematici

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Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ - Δ p u + | u | p - 2 u = f 1 λ ( x ) | u | q - 2 u + 2 α / ( α + β ) g μ | u | α - 2 u | v | β , x ∈ Ω, ⎨ - Δ p v + | v | p - 2 v = f 2 λ ( x ) | v | q - 2 v + 2 β / ( α + β ) g μ | u | α | v | β - 2 v , x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and Ω N is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and f i λ i ( x ) = λ i f i + ( x ) + f i - ( x ) (i = 1,2) are sign-changing functions, where f i ± ( x ) = m a x ± f i ( x ) , 0 , g μ ( x ) = a ( x ) + μ b ( x ) , and Δ p u = d i v ( | u | p - 2 u ) denotes the p-Laplace operator. We use variational methods.

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

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We study the existence of positive solutions to ⎧ d i v ( | u | p - 2 u ) + q ( x ) u - γ = 0 on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is N or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

Existence and nonexistence of solutions for a quasilinear elliptic system

Qin Li, Zuodong Yang (2015)

Annales Polonici Mathematici

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By a sub-super solution argument, we study the existence of positive solutions for the system ⎧ - Δ p u = a ( x ) F ( x , u , v ) in Ω, ⎪ - Δ q v = a ( x ) F ( x , u , v ) in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in N with smooth boundary or Ω = N . A nonexistence result is obtained for radially symmetric solutions.

Finite element variational crimes in the case of semiregular elements

Alexander Ženíšek (1996)

Applications of Mathematics

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The finite element method for a strongly elliptic mixed boundary value problem is analyzed in the domain Ω whose boundary Ω is formed by two circles Γ 1 , Γ 2 with the same center S 0 and radii R 1 , R 2 = R 1 + ϱ , where ϱ R 1 . On one circle the homogeneous Dirichlet boundary condition and on the other one the nonhomogeneous Neumann boundary condition are prescribed. Both possibilities for u = 0 are considered. The standard finite elements satisfying the minimum angle condition are in this case inconvenient; thus...

Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

Djairo Guedes de Figueiredo, Jean-Pierre Gossez, Pedro Ubilla (2006)

Journal of the European Mathematical Society

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We study the existence, nonexistence and multiplicity of positive solutions for the family of problems Δ u = f λ ( x , u ) , u H 0 1 ( Ω ) , where Ω is a bounded domain in N , N 3 and λ > 0 is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely λ a ( x ) u q + b ( x ) u p , where 0 q < 1 < p 2 * 1 . The coefficient a ( x ) is assumed to be nonnegative but b ( x ) is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential...

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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We study the chemotaxis system with singular sensitivity and logistic-type source: u t = Δ u - χ · ( u v / v ) + r u - μ u k , 0 = Δ v - v + u under the non-flux boundary conditions in a smooth bounded domain Ω n , χ , r , μ > 0 , k > 1 and n 1 . It is shown with k ( 1 , 2 ) that the system possesses a global generalized solution for n 2 which is bounded when χ > 0 is suitably small related to r > 0 and the initial datum is properly small, and a global bounded classical solution for n = 1 .

Asymptotic analysis and sign-changing bubble towers for Lane–Emden problems

Francesca De Marchis, Isabella Ianni, Filomena Pacella (2015)

Journal of the European Mathematical Society

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We consider the semilinear Lane–Emden problem where p > 1 and Ω is a smooth bounded domain of 2 . The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ( p ) , as p + . Among other results we show, under some symmetry assumptions on Ω , that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as p + , and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of...

On annealed elliptic Green's function estimates

Daniel Marahrens, Felix Otto (2015)

Mathematica Bohemica

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We consider a random, uniformly elliptic coefficient field a on the lattice d . The distribution · of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green’s function G ( t , x , y ) satisfy optimal annealed estimates which are L 2 and L 1 , respectively, in probability, i.e., they obtained bounds on | x G ( t , x , y ) | 2 1 / 2 and | x y G ( t , x , y ) | . In particular, the elliptic Green’s function G ( x , y ) satisfies optimal annealed bounds. In their recent work,...

Perturbed nonlinear degenerate problems in N

A. El Khalil, S. El Manouni, M. Ouanan (2009)

Applicationes Mathematicae

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ d i v ( x , u ) + a ( x ) | u | p - 2 u = g ( x ) | u | p - 2 u + h ( x ) | u | s - 1 u in N ⎨ ⎩ u > 0, l i m | x | u ( x ) = 0 , where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

T-p(x)-solutions for nonlinear elliptic equations with an L¹-dual datum

El Houssine Azroul, Abdelkrim Barbara, Meryem El Lekhlifi, Mohamed Rhoudaf (2012)

Applicationes Mathematicae

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We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem - d i v ( a ( x , u , u ) ) + g ( x , u ) = f - d i v F in Ω, where Ω is a bounded open domain of N , N ≥ 2 and a : Ω × × N N is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to i = 1 N L p ' ( · ) ( Ω ) .

Partially elliptic differential equations having distributions as their right members

H. Marcinkowska

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ContentsIntroduction.............................................................................................................................31. Definitions, notations and some auxiliary lemmas...................................................42. The definition of the spaces H p , q ; Y ( Ω , ) ..........................................................73. Some properties of the spaces H p , q ; Y ( Ω , ) ...................................................104. Some examples of the spaces H p , q ; Y ( Ω , ) ....................................................155....

Existence of renormalized solutions for some degenerate and non-coercive elliptic equations

Youssef Akdim, Mohammed Belayachi, Hassane Hjiaj (2023)

Mathematica Bohemica

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This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by t 2 - div ( b ( | u | ) | u | p - 2 u ) + d ( | u | ) | u | p = f - div ( c ( x ) | u | α ) in Ω , u = 0 on Ω , t where Ω is a bounded open set of N ( N 2 ) with 1 < p < N and f L 1 ( Ω ) , under some growth conditions on the function b ( · ) and d ( · ) , where c ( · ) is assumed to be in L N ( p - 1 ) ( Ω ) . We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

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We consider the Yamabe type family of problems ( P ε ) : Δ u ε = u ε ( n + 2 ) / ( n 2 ) , u ε > 0 in A ε , u ε = 0 on A ε , where A ε is an annulus-shaped domain of n , n 3 , which becomes thinner as ε 0 . We show that for every solution u ε , the energy A ε | u | 2 as well as the Morse index tend to infinity as ε 0 . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on n , a half-space or an infinite strip. Our argument also involves a Liouville...

The Mordell-Weil bases for the elliptic curve y 2 = x 3 - m 2 x + m 2

Sudhansu Sekhar Rout, Abhishek Juyal (2021)

Czechoslovak Mathematical Journal

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Let D m be an elliptic curve over of the form y 2 = x 3 - m 2 x + m 2 , where m is an integer. In this paper we prove that the two points P - 1 = ( - m , m ) and P 0 = ( 0 , m ) on D m can be extended to a basis for D m ( ) under certain conditions described explicitly.

Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta, Laurent Véron (2013)

Journal of the European Mathematical Society

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For 0 < p - 1 < q and either ϵ = 1 or ϵ = - 1 , we prove the existence of solutions of - Δ p u = ϵ u q in a cone C S , with vertex 0 and opening S , vanishing on C S , of the form u ( x ) = x - β ω ( x / x ) . The problem reduces to a quasilinear elliptic equation on S and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.