A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities

We consider a nonlinear parametric Dirichlet problem driven by the anisotropic p-Laplacian with the combined effects of “concave” and “convex” terms. The “superlinear” nonlinearity need not satisfy the Ambrosetti-Rabinowitz condition. Using variational methods based on the critical point theory and the Ekeland variational principle, we show that for small values of the parameter, the problem has at least two nontrivial smooth positive solutions.     

Integral representation of random set-valued measures

Abstract

We consider random set-valued measures with values in a separable Banach space. We prove two integral representation theorems using measurable multifunctions and set-valued integrals. The first theorem is valid for all separable Banach spaces, while the second holds for reflexive separable Banach spaces.     

A geometric criterion for compactness of invariant subspaces

Let M be a non-compact homogeneous Riemannian manifold, and let Ω be a compact subgroup of isometries of M. We show, under general conditions, that the Ω-invariant subspace A _Ω of a normed vector space ${A\hookrightarrow L^q(M)}$ A ? L q ( M ) is compactly embedded into L ^q (M) if and only if the group Ω has no orbits with a uniformly bounded diameter in a neighborhood of infinity.     

Existence and uniqueness of positive solutions for the Neumann p-Laplacian

We consider a nonlinear Neumann problem driven by the p-Laplacian and with a Carathéodory reaction which satisfies only a unilateral growth restriction. Using the principal eigenvalue of an eigenvalue problem involving the Neumann p-Laplacian plus an indefinite potential, we produce necessary and sufficient conditions for the existence and uniqueness of positive smooth solutions.     

On the Interplay of Regularity and Decay in Case of Radial Functions II. Homogeneous Spaces

We deal with decay and boundedness properties of elements of radial subspaces of homogeneous Besov and Triebel-Lizorkin spaces. For the region of parameters which are of interest for us these homogeneous spaces are larger than the inhomogeneous counterparts. By switching from the inhomogeneous spaces to the homogeneous classes the properties of the radial elements change. Our investigations are based on the atomic decompositions for radial subspaces in the sense of Epperson and Frazier (J.?Fourier Anal Appl. 1:311?C353, 1995). Finally, we apply these results for deriving some assertions on compact embeddings on unbounded domains.     

Neumann problems resonant at zero and infinity

We consider a semilinear Neumann problem with a reaction which is resonant at both zero and ±∞. Using a combination of methods from critical point theory, together with truncation techniques, the use of upper–lower solutions and of the Morse theory (critical groups), we show that the problem has at least five nontrivial smooth solutions, four of which have constant sign (two positive and two negative).     

Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains \(\Omega \) obtained by different approaches. In particular, we consider three settings \(\mathcal {M}_{u,p}(\Omega )\), \(\mathbb {M}_{u,p}(\Omega )\) and \(\mathfrak {M}_{u,p}(\Omega )\), where \(0<p\le u<\infty \), commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes \(\mathfrak {E}_{\mathsf {G}}(\mathcal {M}_{u,p}(\Omega ))\) as well as \(\mathfrak {E}_{\mathsf {G}}(\mathfrak {M}_{u,p}(\Omega ))\), and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether \(\frac{n}{u}\ge \frac{1}{p}\) or \(\frac{n}{u} < \frac{1}{p}\) plays a decisive role when it comes to the behaviour of these spaces.     

A note on the Σ1 collection scheme and fragments of bounded arithmetic

We show that for each n ≥ 1, if T₂ⁿ does not prove the weak pigeonhole principle for Σ^b_n functions, then the collection scheme B Σ₁ is not finitely axiomatizable over T₂ⁿ. The same result holds with Sⁿ₂ in place of T 2ⁿ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydrophilic-lipophilic balance and surface film state of nonionic surfactants

Summary The polarographic study on the effect of nonionic surfactants on the damping of oxygen maximum confirms the validity ofVan Voorst Vader's rule and a major part which play the hydrophilic-lipophilic balance of surfactant molecules in surface phenomena. On this basis the simple relationships are presented between the molecular area in surface film (A) andGriffin's andDavies' HLB. It is shown that at a specific pressure theA increases with increasing HLB values for homologous series of nonionic surfactants. The limitations of the validity of the relations derived with respect to the specific surface pressure chosen are also considered.

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Zusammenfassung Polarographische Studien über die Wirkung nichtionogener Tenside auf die Dämpfung des Sauerstoff maximums bestätigen die Gültigkeit der Regel vonVan Voorst Vader und den überwiegenden Einfluß, welchen der Anteil hydrophil-lipophil in den Tensidmolekülen für Grenzflächenphänomene spielt. Auf dieser Basis wird eine einfache Beziehung erhalten zwischen dem Flächenbedarf eines Moleküls im Grenzflächenfilm (A) undGriffin's undDavies HLB-Werten. Es wird gezeigt, daß bei einem spezifischen Oberflächendruck dieA-Werte mit steigenden HLB-Werten in homologen Reihen nichtionogener Tenside ansteigen. Die Grenzen der Gültigkeit dieser Regel werden diskutiert.

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With 2 tables     

The decay of theT=1 isospin triplet inA=12 systems: II. The induced interactions in the anisotropy of beta-rays

Nuclear Structure: calculation of the asymmetryα _? ofβ rays emitted from polarized¹²B and¹²N nuclei. Comparison with experimental values. Formulae for the asymmetry parametersα _? ofβ rays emitted from polarized nuclei is derived using the formalism developed by Behrens-Bühring. The induced interactions are studied in detail. It is demonstrated that the expressions forα _? are independent of the induced pseudoscalar interaction. It was found that the corrections to the lowest order approximation are small. Experimental data of the Louvain and Osaka laboratories are analyzed. We should point out that the data of the asymmetry of theβ ^? decay of¹²B obtained in these two laboratories are quite different. Although the data reported by the Osaka Group require the existence of the induced pseudotensor (IPT) interaction as it has also been stated by other authors, if we combine data of¹²N from Osaka and of¹²B from Lauvain such experimental results can be understood without including the IPT interaction. Thus, we found some evidence that the IPT interaction could be not necessary to explain the asymmetries in the decay of members of theA=12 system.