Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations

Leszek Gęba; Tadeusz Pruszko

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Displaying similar documents to “Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations”

Positive solutions of nonlinear elliptic systems

Robert Dalmasso (1993)

Annales Polonici Mathematici

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We study the existence and nonexistence of positive solutions of nonlinear elliptic systems in an annulus with Dirichlet boundary conditions. In particular, $L^{\infty}$ a priori bounds are obtained. We also study a general multiple linear eigenvalue problem on a bounded domain.

Difference method for an elliptic system of nonlinear differential-functional equations.

Mosurski, Ryszard (2005)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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Three methods for the study of semilinear equations at resonance

Bogdan Przeradzki (1993)

Colloquium Mathematicae

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Three methods for the study of the solvability of semilinear equations with noninvertible linear parts are compared: the alternative method, the continuation method of Mawhin and a new perturbation method [22]-[27]. Some extension of the last method and applications to differential equations in Banach spaces are presented.

On the solvability of nonlinear elliptic equations in Sobolev spaces

Piotr Fijałkowski (1992)

Annales Polonici Mathematici

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We consider the existence of solutions of the system (*) $P (D) u^{l} = F (x, (\partial^{α} u))$ , l = 1,...,k, $x \in ℝ^{n}$ $(u = (u ¹, . . ., u^{k}))$ in Sobolev spaces, where P is a positive elliptic polynomial and F is nonlinear.

On the restricted Waring problem over $_{2^{n}} [t]$

Luis Gallardo (2000)

Acta Arithmetica

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1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, $_{16}$ , each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is...

A remark on product of Dirichlet L-functions

Kirti Joshi, C. S. Yogananda (1999)

Acta Arithmetica

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While trying to understand the methods and the results of [3], especially in Section 2, we stumbled on an identity (*) below, which looked worth recording since we could not locate it in the literature. We would like to thank Dinesh Thakur and Dipendra Prasad for their comments.

Minimax inequality for a special class of functionals and its application to existence of three solutions for a Dirichlet problem in one-dimensional case.

Afrouzi, G.A., Heidarkhani, S., Hossienzadeh, H., Yazdani, A. (2010)

The Journal of Nonlinear Sciences and its Applications

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Existence problems for $ω$ -closed orbits.

Avramescu, Cezar (2002)

Analele Ştiinţifice ale Universităţii “Ovidius" Constanţa. Seria: Matematică

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Reverse functional inequalities and their applications to nonlinear elliptic boundary value problems.

Klimov, V.S., Pavlenko, A.N. (2001)

Sibirskij Matematicheskij Zhurnal

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On a discrete version of the antipodal theorem

Krzysztof Oleszkiewicz (1996)

Fundamenta Mathematicae

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The classical theorem of Borsuk and Ulam [2] says that for any continuous mapping $f : S^{k} \to ℝ^{k}$ there exists a point $x \in S^{k}$ such that f(-x) = f(x). In this note a discrete version of the antipodal theorem is proved in which $S^{k}$ is replaced by the set of vertices of a high-dimensional cube equipped with Hamming’s metric. In place of equality we obtain some optimal estimates of $i n f_{x} | | f (x) - f (- x) | |$ which were previously known (as far as the author knows) only for f linear (cf. [1]).

Qualitative behavior of a class of second order nonlinear differential equations on the halfline

Svatoslav Staněk (1993)

Annales Polonici Mathematici

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A differential equation of the form (q(t)k(u)u')' = F(t,u)u' is considered and solutions u with u(0) = 0 are studied on the halfline [0,∞). Theorems about the existence, uniqueness, boundedness and dependence of solutions on a parameter are given.

On the exponent of the ideal class groups of imaginary extensions of $_{q} (x)$

Hershy Kisilevsky, Francesco Pappalardi (1995)

Acta Arithmetica

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Growth of the product $\prod_{j = 1}^{n} (1 - x^{a_{j}})$

J. P. Bell, P. B. Borwein, L. B. Richmond (1998)

Acta Arithmetica

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We estimate the maximum of $\prod_{j = 1}^{n} | 1 - x^{a_{j}} |$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_{j}$ is $j^{k}$ or when $a_{j}$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_{j}$ is j. In contrast we show, under fairly general conditions, that the maximum is less than $2^{n} / n^{r}$ , where r is an arbitrary positive number. One consequence...

Ergodic averages and free $ℤ^{2}$ actions

Zoltán Buczolich (1999)

Fundamenta Mathematicae

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If the ergodic transformations S, T generate a free $ℤ^{2}$ action on a finite non-atomic measure space (X,S,µ) then for any $c_{1}, c_{2} \in ℝ$ there exists a measurable function f on X for which ${(N + 1)}^{- 1} \sum_{j = 0}^{N} f (S^{j} x) \to c_{1}$ and ${(N + 1)}^{- 1} \sum_{j = 0}^{N} f (T^{j} x) \to c_{2} µ$ -almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

Strongly meager sets and subsets of the plane

Janusz Pawlikowski (1998)

Fundamenta Mathematicae

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Let $X \subseteq 2^{w}$ . Consider the class of all Borel $F \subseteq X \times 2^{w}$ with null vertical sections $F_{x}$ , x ∈ X. We show that if for all such F and all null Z ⊆ X, $\cup_{x \in Z} F_{x}$ is null, then for all such F, $\cup_{x \in X} F_{x} \neq 2^{w}$ . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P].

Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

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For X ⊆ [0,1], let $D_{X}$ denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of $D_{X}$ . For α ≥ 3, X is properly $D_{ξ} (Π_{α}^{0})$ iff $D_{X}$ is properly $D_{ξ} (Π_{1 + α}^{0})$ . We also show that for every nonempty set X ⊆[0,1], $D_{X}$ is $Π_{3}^{0}$ -hard. For each nonempty $Π_{2}^{0}$ set X ⊆ [0,1], in particular for X = x, $D_{X}$ is $Π_{3}^{0}$ -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is $Π_{3}^{0}$ -complete. Moreover,...