Multiple solutions to logarithmic Schrodinger equations with periodic potential

We study a class of logarithmic Schrodinger equations with periodic potential which come from physically relevant situations and obtain the existence of infinitely many geometrically distinct solutions.Comment: 3 pages, corrigendum to version

Ground states of critical and supercritical problems of Brezis-Nirenberg type

We study the existence of symmetric ground states to the supercritical problem $$-\Delta v=\lambda v+\left\vert v\right\vert ^{p-2}v\text{ \ in }\Omega,\qquad v=0\text{ on }\partial\Omega,$$ in a domain of the form $$\Omega=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert y\right\vert ,z\right) \in\Theta\},$$ where $\Theta$ is a bounded smooth domain such that $\overline{\Theta} \subset\left( 0,\infty\right) \times\mathbb{R}^{N-k-1},$ $1\leq k\leq N-3,$ $\lambda\in\mathbb{R},$ and $p=\frac{2(N-k)}{N-k-2}$ is the $(k+1)$-st critical exponent. We show that symmetric ground states exist for $\lambda$ in some interval to the left of each symmetric eigenvalue, and that no symmetric ground states exist in some interval $(-\infty,\lambda_{\ast})$ with $\lambda_{\ast}>0$ if $k\geq2.$ Related to this question is the existence of ground states to the anisotropic critical problem $$-\text{div}(a(x)\nabla u)=\lambda b(x)u+c(x)\left\vert u\right\vert ^{2^{\ast }-2}u\quad\text{in}\ \Theta,\qquad u=0\quad\text{on}\ \partial\Theta,$$ where $a,b,c$ are positive continuous functions on $\overline{\Theta}.$ We give a minimax characterization for the ground states of this problem, study the ground state energy level as a function of $\lambda,$ and obtain a bifurcation result for ground states

Mechanisms of Inequality: Unequal Access to Organizational Power and the Gender Wage Gap

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Ethnic Segregation and Educational Outcomes in Swedish Comprehensive Schools

We ask whether ethnic density in Swedish comprehensive schools affect teacher-assigned school grades in ninth grade (age 16). The data, based on two entire cohorts who graduated in 1998 and 1999 (188,000 pupils and 1,043 schools), link school information with Census data on social origin, and enable us to distinguish first- from second generation immigrants. Using multilevel analysis we find the proportion of first, but not the second, generation immigrant pupils in a school to depress grades in general, but particularly for (first generation) immigrant pupils. Passing a threshold of more than 40 percent immigrants reduces grades with around a fifth of a standard deviation, affecting fourteen percent of immigrant children. Our main results are robust to model specifications which address omitted variable bias both at individual- and school-level. One policy implication of our results is that desegregation policies which concentrated on the two per cent most segregated schools would probably improve school results and reduce ethnic inequality.Ethnic inequality; Immigrant schooling; Educational attainment; Contextual effects; Ethnic inequality; Immigrant schooling