Existence results of positive solutions for Kirchhoff type equations via bifurcation methods

In this paper we address the following Kirchhoff type problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta(g(|\nabla u|_2^2) u + u^r) = a u + b u^p& \mbox{in}~\Omega, u>0& \mbox{in}~\Omega, u= 0& \mbox{on}~\partial\Omega, \end{array} \right. \end{equation*} in a bounded and smooth domain $\Omega$ in ${\rm I}\hskip -0.85mm{\rm R}$. By using change of variables and bifurcation methods, we show, under suitable conditions on the parameters $a,b,p,r$ and the nonlinearity $g$, the existence of positive solutions.Comment: 18 pages, 1 figur

Biodiversidade, população e economia: uma região de mata atlântica [Biodiversity, Population, and Economy: a region of atlantic forest]

biodiversity; Minas Gerais; Rio Doce; mata atlântica; atlantic forest; sustainable development; conservation; natural resources; environment; environmental economics

Biodiversidade, população e economia: uma região de mata atlântica [Biodiversity, Population, and Economy: a region of atlantic forest]

Minas Gerais; Rio Doce; mata atlântica; atlantic forest; sustainable development; conservation; regional development; environment

Beachrocks of the northeast of Brazil: local effects of sea level fluctuations in a far-field during in Holocene

Along the coast of the state of Pernambuco, NE Brazil, the presence of coastal outcrops aligned with the shoreline and distributed above and below the mean sea level is remarkable. These outcrops were sampled, and petrological, isotopic, and geochronological analyses were integrated to investigate the local effects of the Holocene sea-level fluctuations. The results indicate that the rocks are recent (< 8ka), formed by cementation of beach sands (magnesian calcite cement, predominance of quartz and presence of marine bioclasts, signs of compaction, fractures, and dissolution), in a tropical climate and warm waters, which allows to classify them as typical examples of beachrocks. The carbonate cement consisted of Mg-rich calcite with values ranging from −1.1‰ to 3.5‰ for δ13C and from −0.9‰ to 0.5‰ for δ18OVPDB. These isotopic values are typical of marine carbonate cemented deposits in shallow marine environments under freshwater influence (meteoric vadose environment). In these conditions, the cementing processes occurred in the intertidal zones, which reinforce the use of these data as indicators of ancient sea levels. The sea level fluctuations could be divided, in chronological order, into three different phases, starting with a rapid sea level rise followed by a relative stabilization at a maximum level and, finally, a decreasing phase. Recent sea level fluctuations in Pernambuco are represented by beachrocks with ages between 8 ka and 800 years and a maximum level of 2 to 3 meters above the present mean level. In scenarios of a near future with a general global sea level rise pattern, significant local geomorphological changes may occur, which means a great future challenge to society

Biodiversidade, população e economia: uma região de mata atlântica [Biodiversity, Population, and Economy: a region of atlantic forest]

Minas Gerais; Rio Doce; mata atlântica; atlantic forest; sustainable development; conservation; regional development; environment