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Indefinite Quasilinear Neumann Problem on Unbounded Domains

J. Chabrowski (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

We investigate the solvability of the quasilinear Neumann problem (1.1) with sub- and supercritical exponents in an unbounded domain Ω. Under some integrability conditions on the coefficients we establish embedding theorems of weighted Sobolev spaces into weighted Lebesgue spaces. This is used to obtain solutions through a global minimization of a variational functional.

Inégalités de résolvante pour l’opérateur de Schrödinger avec potentiel multipolaire critique

Thomas Duyckaerts (2006)

Bulletin de la Société Mathématique de France

On étudie un opérateur de la forme - Δ + V sur d , où V est un potentiel admettant plusieurs pôles en a / r 2 . Plus précisément, on démontre l’estimation de résolvante tronquée à hautes fréquences, classique dans les cas non-captifs, et qui implique l’effet régularisant standard pour l’équation de Schrödinger correspondante. La preuve est basée sur l’introduction d’une mesure de défaut micro-locale semi-classique. On démontre également, dans le même contexte, des inégalités de Strichartz pour l’équation de Schrödinger....

Inequalities of Korn's type, uniform with respect to a class of domains

Ivan Hlaváček (1989)

Aplikace matematiky

Inequalities of Korn's type involve a positive constant, which depends on the domain, in general. A question arises, whether the constants possess a positive infimum, if a class of bounded two-dimensional domains with Lipschitz boundary is considered. The proof of a positive answer to this question is shown for several types of boundary conditions and for two classes of domains.

Infinitely many positive solutions for the Neumann problem involving the p-Laplacian

Giovanni Anello, Giuseppe Cordaro (2003)

Colloquium Mathematicae

We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ - Δ p u + λ ( x ) | u | p - 2 u = μ f ( x , u ) in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where Ω N is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, λ L ( Ω ) with e s s i n f x Ω λ ( x ) > 0 and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

Infinitely many solutions for a class of semilinear elliptic equations in R N

Francesca Alessio, Paolo Caldiroli, Piero Montecchiari (2001)

Bollettino dell'Unione Matematica Italiana

Si considera una classe di equazioni ellittiche semilineari su R N della forma - Δ u + u = a x u p - 1 u con p > 1 sottocritico (o con nonlinearità più generali) e a x funzione limitata. In questo articolo viene presentato un risultato di genericità sull'esistenza di infinite soluzioni, rispetto alla classe di coefficienti a x limitati su R N e non negativi all'infinito.

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