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It is established the existence of positive least energy solution for the following class of planar elliptic systems in the zero mass case - Δ u - Δ q u + ϕ | u | r - 2 u = λ | u | p - 2 u , in R 2 , Δ ϕ = 2 π | u | r , & in R 2 , where λ ≥ 0 , 1 < q < 2 , q ∗ : = 2 q / (2 - q) < r < ∞ and p ≥ 2 r . Due to the nature of the problem, we deal with the logarithmic integral kernel. Our approach is based on Nehari manifold and a version of the Principle of Symmetric Criticality due to Palais. Furthermore, we study the asymptotic behavior of the solutions whenever the parameter λ goes to zero or infinity. Finally, we study regularity of solutions applying Moser iteration scheme. [ABSTRACT FROM AUTHOR]

In this work, we establish some abstract results on the perspective of the fractional Musielak–Sobolev spaces, such as: uniform convexity, Radon–Riesz property with respect to the modular function, (S +) -property, Brezis–Lieb type Lemma to the modular function and monotonicity results. Moreover, we apply the theory developed to study the existence of solutions to the following class of nonlocal problems (- Δ) Φ x , y s u = f (x , u) , in Ω , u = 0 , on R N \ Ω , where N ≥ 2 , Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω and f : Ω × R → R is a Carathéodory function not necessarily satisfying the Ambrosetti–Rabinowitz condition. Such class of problems enables the presence of many particular operators, for instance, the fractional operator with variable exponent, double-phase and double-phase with variable exponent operators, anisotropic fractional p-Laplacian, among others. [ABSTRACT FROM AUTHOR]

It is establish regularity results for weak solutions of quasilinear elliptic problems driven by the well known Φ -Laplacian operator given by - Δ Φ u = g (x , u) , in Ω , u = 0 , on ∂ Ω , where Δ Φ u : = div (ϕ (| ∇ u |) ∇ u) and Ω ⊂ R N , N ≥ 2 , is a bounded domain with smooth boundary ∂ Ω . Our work concerns on nonlinearities g which can be homogeneous or non-homogeneous. For the homogeneous case we consider an existence result together with a regularity result proving that any weak solution remains bounded. Furthermore, for the non-homogeneous case, the nonlinear term g can be subcritical or critical proving also that any weak solution is bounded. The proofs are based on Moser's iteration in Orlicz and Orlicz-Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this work, we studied the optical absorption spectral differences between doped and slightly doped polyaniline films and a derivative by photopyroelectric and photoacoustic spectroscopies. There exist great spectral differences between doped and slightly doped samples as shown by conventional optical absorption spectroscopy, but not greatly evidenced by photopyroelectric spectroscopy. The latter was applied to obtain thermal parameters such as thermal diffusivity and thermal conductivity, and as a result it showed that these thermal properties of polyaniline films are very similar for doped and slightly doped samples. Also it showed that for thicker films (about 20 μm), there are no significant optical absorption differences between them. However, for thin films both techniques showed greater optical absorption differences for doped and slightly doped samples, mainly detected by photoacoustic spectroscopy. These behaviors are in accordance with published results which is the granular metal model for polyaniline. This model explains the polyaniline polymeric matrix as formed by conductive islands in the insulating bulk material. [ABSTRACT FROM AUTHOR]

In this study, we carried out a quantitative depth profile analysis of the optical properties of slightly and strongly doped polyaniline (PANI) films, using both photoacoustic and photopyroelectric spectroscopies. As the thermal diffusion length varies inversely to chopper modulation frequency, then at high chopping frequencies information about the sample near the surface is obtained, while at low chopping frequencies the data come from deeper within the sample. The objective of this study is to analyze how the polyaniline doping process in acidic media occurs in the polymeric chain using the photothermal techniques. The results showed that polymeric chains of the doped samples are composed of undoped regions, while polymeric chains of slightly doped samples present doped regions to a lesser degree. These conclusions are in accordance with published results in which doped PANI films exhibit a granular conductive regions in a quasi undoped matrix. In addition, the photothermal spectra showed that the doping process of PANI, in which films are dipped in hydrochloric acid solution, is more effective in the external surface of the films. [ABSTRACT FROM AUTHOR]