Displaying similar documents to “Existence of weak solutions for elliptic Dirichlet problems with variable exponent”

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 > 0 in Ω ε ω , u = 0 on Ω ε ω , where Ω and ω are smooth bounded domains in N , N 3 , ε > 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...

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.

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...

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.

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.

Nonlinear fourth order problems with asymptotically linear nonlinearities

Abir Amor Ben Ali, Makkia Dammak (2024)

Mathematica Bohemica

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We investigate some nonlinear elliptic problems of the form Δ 2 v + σ ( x ) v = h ( x , v ) in Ω , v = Δ v = 0 on Ω , ( P ) where Ω is a regular bounded domain in N , N 2 , σ ( x ) a positive function in L ( Ω ) , and the nonlinearity h ( x , t ) is indefinite. We prove the existence of solutions to the problem (P) when the function h ( x , t ) is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

Quasilinear elliptic equations with discontinuous coefficients

Lucio Boccardo, Giuseppe Buttazzo (1988)

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

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We prove an existence result for equations of the form { - D i ( a i j ( x , u ) D j u ) = f in Ω u H 0 1 ( Ω ) . where the coefficients a i j ( x , s ) satisfy the usual ellipticity conditions and hypotheses weaker than the continuity with respect to the variable s . Moreover, we give a counterexample which shows that the problem above may have no solution if the coefficients a i j ( x , s ) are supposed only Borel functions

On Kneser solutions of the n -th order nonlinear differential inclusions

Martina Pavlačková (2019)

Czechoslovak Mathematical Journal

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The paper deals with the existence of a Kneser solution of the n -th order nonlinear differential inclusion x ( n ) ( t ) - A 1 ( t , x ( t ) , ... , x ( n - 1 ) ( t ) ) x ( n - 1 ) ( t ) - ... - A n ( t , x ( t ) , ... , x ( n - 1 ) ( t ) ) x ( t ) for a.a. t [ a , ) , where a ( 0 , ) , and A i : [ a , ) × n , i = 1 , ... , n , are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

On behavior of solutions to a chemotaxis system with a nonlinear sensitivity function

Senba, Takasi, Fujie, Kentarou

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In this paper, we consider solutions to the following chemotaxis system with general sensitivity τ u t = Δ u - · ( u χ ( v ) ) in Ω × ( 0 , ) , η v t = Δ v - v + u in Ω × ( 0 , ) , u ν = u ν = 0 on Ω × ( 0 , ) . Here, τ and η are positive constants, χ is a smooth function on ( 0 , ) satisfying χ ' ( · ) > 0 and Ω is a bounded domain of 𝐑 n ( n 2 ). It is well known that the chemotaxis system with direct sensitivity ( χ ( v ) = χ 0 v , χ 0 > 0 ) has blowup solutions in the case where n 2 . On the other hand, in the case where χ ( v ) = χ 0 log v with 0 < χ 0 1 , any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness...

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 .

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

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Let p be an odd prime, and let a be an integer not divisible by p . When m is a positive integer with p 1 ( mod 2 m ) and 2 is an m th power residue modulo p , we determine the value of the product k R m ( p ) ( 1 + tan ( π a k / p ) ) , where R m ( p ) = { 0 < k < p : k is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y , then k R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

Traceability in { K 1 , 4 , K 1 , 4 + e } -free graphs

Wei Zheng, Ligong Wang (2019)

Czechoslovak Mathematical Journal

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A graph G is called { H 1 , H 2 , , H k } -free if G contains no induced subgraph isomorphic to any graph H i , 1 i k . We define σ k = min i = 1 k d ( v i ) : { v 1 , , v k } is an independent set of vertices in G . In this paper, we prove that (1) if G is a connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and σ 3 ( G ) n - 1 , then G is traceable, (2) if G is a 2-connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and | N ( x 1 ) N ( x 2 ) | + | N ( y 1 ) N ( y 2 ) | n - 1 for any two distinct pairs of non-adjacent vertices { x 1 , x 2 } , { y 1 , y 2 } of G , then G is traceable, i.e., G has a Hamilton path, where K 1 , 4 + e is a graph obtained by joining a pair of non-adjacent vertices in a K 1 , 4 .

Asymptotic properties of a ϕ -Laplacian and Rayleigh quotient

Waldo Arriagada, Jorge Huentutripay (2020)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider the ϕ -Laplacian problem with Dirichlet boundary condition, - div ϕ ( | u | ) u | u | = λ g ( · ) ϕ ( u ) in Ω , λ and u | Ω = 0 . The term ϕ is a real odd and increasing homeomorphism, g is a nonnegative function in L ( Ω ) and Ω N is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both...