Displaying similar documents to “Infinitely many solutions for a semilinear elliptic equation in N via a perturbation method”

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.

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.

Nonrectifiable oscillatory solutions of second order linear differential equations

Takanao Kanemitsu, Satoshi Tanaka (2017)

Archivum Mathematicum

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The second order linear differential equation ( p ( x ) y ' ) ' + q ( x ) y = 0 , x ( 0 , x 0 ] is considered, where p , q C 1 ( 0 , x 0 ] , p ( x ) > 0 , q ( x ) > 0 for x ( 0 , x 0 ] . Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near x = 0 without the Hartman–Wintner condition.

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...

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...

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Olivier Rey, Juncheng Wei (2005)

Journal of the European Mathematical Society

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We show that the critical nonlinear elliptic Neumann problem Δ u μ u + u 7 / 3 = 0 in Ω , u > 0 in Ω , u ν = 0 on Ω , where Ω is a bounded and smooth domain in 5 , has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K , there exists μ K > 0 such that for 0 < μ < μ K , the above problem has a nontrivial solution which blows up at K interior points in Ω , as μ 0 . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

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.

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator Δ 2 in a bounded domain Ω n . We state some existence results when n 8 . Moreover, we consider 5 n 7 , expecially when Ω is a ball in n .

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....

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,...

L p type mapping estimates for oscillatory integrals in higher dimensions

G. Sampson (2006)

Studia Mathematica

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We show in two dimensions that if K f = ² k ( x , y ) f ( y ) d y , k ( x , y ) = ( e i x a · y b ) / ( | x - y | η ) , p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), v p ( y ) = y ( p / p ' ) ( 1 ̅ - b / a ) , then | | K f | | p C | | f | | p , v p if η + α₁ + α₂ < 2, α j = 1 - b j / a j , j = 1,2. Our methods apply in all dimensions and also for more general kernels.

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.

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 ) .

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...

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.

On some L p -estimates for solutions of elliptic equations in unbounded domains

Sara Monsurrò, Maria Transirico (2015)

Mathematica Bohemica

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In this review article we present an overview on some a priori estimates in L p , p > 1 , recently obtained in the framework of the study of a certain kind of Dirichlet problem in unbounded domains. More precisely, we consider a linear uniformly elliptic second order differential operator in divergence form with bounded leading coeffcients and with lower order terms coefficients belonging to certain Morrey type spaces. Under suitable assumptions on the data, we first show two L p -bounds, p > 2 , for...

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

In this paper we consider a nonlinear elliptic equation with critical growth for the operator Δ 2 in a bounded domain Ω n . We state some existence results when n 8 . Moreover, we consider 5 n 7 , expecially when Ω is a ball in n .

Existence of positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

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We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ l i m | x | u ( x ) = 0 , where Ω = x N : | x | > r , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and 0 < r r K ( r ) d r < , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the...

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).