Displaying similar documents to “Quasilinear elliptic equations with discontinuous coefficients”

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.

An Elliptic Problem with a Lower Order Term Having Singular Behaviour

Daniela Giachetti, François Murat (2009)

Bollettino dell'Unione Matematica Italiana

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We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution u in a singular way and on its gradient D u with quadratic growth. The prototype of the problem under consideration is { - Δ u + λ u = ± | D u | 2 | u | k + f in Ω , u = 0 on Ω , where λ > 0 , k > 0 ; f ( x ) L ( Ω ) , f ( x ) 0 (and so u 0 ). If 0 < k < 1 , we prove the existence of a solution for both the "+" and the "-" signs, while if k 1 , we prove the existence of a solution for the "+" sign only.

Existence of weak solutions for elliptic Dirichlet problems with variable exponent

Sungchol Kim, Dukman Ri (2023)

Mathematica Bohemica

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This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type - div a ( x , u , u ) + b ( x , u , u ) = 0 in Ω , u = 0 on Ω , where Ω is a bounded domain of n , n 2 . In particular, we do not require strict monotonicity of the principal part a ( x , z , · ) , while the approach is based on the variational method and results of the variable exponent function spaces.

Hölder continuity of bounded generalized solutions for some degenerated quasilinear elliptic equations with natural growth terms

Salvatore Bonafede (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation i = 1 m x i a i ( x , u , u ) - c 0 | u | p - 2 u = f ( x , u , u ) , assuming that the principal part of the equation satisfies the following degenerate ellipticity condition λ ( | u | ) i = 1 m a i ( x , u , η ) η i ν ( x ) | η | p , and the lower-order term f has a natural growth with respect to u .

Existence of solutions for some quasilinear p ( x ) -elliptic problem with Hardy potential

Elhoussine Azroul, Mohammed Bouziani, Hassane Hjiaj, Ahmed Youssfi (2019)

Mathematica Bohemica

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We consider the anisotropic quasilinear elliptic Dirichlet problem - i = 1 N D i a i ( x , u , u ) + | u | s ( x ) - 1 u = f + λ | u | p 0 ( x ) - 2 u | x | p 0 ( x ) in Ω , u = 0 on Ω , where Ω is an open bounded subset of N containing the origin. We show the existence of entropy solution for this equation where the data f is assumed to be in L 1 ( Ω ) and λ is a positive constant.

Persistence of Coron’s solution in nearly critical problems

Monica Musso, Angela Pistoia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the problem - Δ u = u N + 2 N - 2 + λ in Ω ε ω , u &gt; 0 in Ω ε ω , u = 0 on Ω ε ω , where Ω and ω are smooth bounded domains in N , N 3 , ε &gt; 0 and λ . We prove that if the size of the hole ε goes to zero and if, simultaneously, the parameter λ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

Positive solutions for concave-convex elliptic problems involving p ( x ) -Laplacian

Makkia Dammak, Abir Amor Ben Ali, Said Taarabti (2022)

Mathematica Bohemica

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We study the existence and nonexistence of positive solutions of the nonlinear equation - Δ p ( x ) u = λ k ( x ) u q ± h ( x ) u r in Ω , u = 0 on Ω where Ω N , N 2 , is a regular bounded open domain in N and the p ( x ) -Laplacian Δ p ( x ) u : = div ( | u | p ( x ) - 2 u ) is introduced for a continuous function p ( x ) > 1 defined on Ω . The positive parameter λ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive...

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...

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.

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.

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,...

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....

Regularity of renormalized solutions to nonlinear elliptic equations away from the support of measure data

Andrea Dall&amp;#039;Aglio, Sergio Segura de León (2019)

Czechoslovak Mathematical Journal

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We prove boundedness and continuity for solutions to the Dirichlet problem for the equation - div ( a ( x , u ) ) = h ( x , u ) + μ , in Ω N , where the left-hand side is a Leray-Lions operator from W 0 1 , p ( Ω ) into W - 1 , p ' ( Ω ) with 1 < p < N , h ( x , s ) is a Carathéodory function which grows like | s | p - 1 and μ is a finite Radon measure. We prove that renormalized solutions, though not globally bounded, are Hölder-continuous far from the support of μ .

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | Ω = 0 , u &gt; 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 &lt; p &lt; , p - 1 &lt; q p * - 1 , λ &gt; 0 , and 0 &lt; δ &lt; 1 . As usual, p * = N p N - p if 1 &lt; p &lt; N , p * ( p , ) is arbitrarily large if p = N , and p * = if p &gt; N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

A Littlewood-Paley type inequality with applications to the elliptic Dirichlet problem

Caroline Sweezy (2007)

Annales Polonici Mathematici

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Let L be a strictly elliptic second order operator on a bounded domain Ω ⊂ ℝⁿ. Let u be a solution to L u = d i v f in Ω, u = 0 on ∂Ω. Sufficient conditions on two measures, μ and ν defined on Ω, are established which imply that the L q ( Ω , d μ ) norm of |∇u| is dominated by the L p ( Ω , d v ) norms of d i v f and | f | . If we replace |∇u| by a local Hölder norm of u, the conditions on μ and ν can be significantly weaker.

On a sequence formed by iterating a divisor operator

Bellaouar Djamel, Boudaoud Abdelmadjid, Özen Özer (2019)

Czechoslovak Mathematical Journal

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Let be the set of positive integers and let s . We denote by d s the arithmetic function given by d s ( n ) = ( d ( n ) ) s , where d ( n ) is the number of positive divisors of n . Moreover, for every , m we denote by δ s , , m ( n ) the sequence d s ( d s ( ... d s ( d s ( n ) + ) + ... ) + ) m -times = d s ( n ) for m = 1 , d s ( d s ( n ) + ) for m = 2 , d s ( d s ( d s ( n ) + ) + ) for m = 3 , We present classical and nonclassical notes on the sequence ( δ s , , m ( n ) ) m 1 , where , n , s are understood as parameters.