On least Energy Solutions to A Semilinear Elliptic Equation in A Strip

We consider the following semilinear elliptic equation on a strip: $$\left\{{array}{l} \Delta u-u + u^p=0 \ {in} \ \R^{N-1} \times (0, L), u>0, \frac{\partial u}{\partial \nu}=0 \ {on} \ \partial (\R^{N-1} \times (0, L)) {array} \right.$$ where $1< p\leq \frac{N+2}{N-2}$. When $1<p 0$ such that for $L \leq L_{*}$, the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for $L >L_{*}$, the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers $L_{*}<L_{**}$ such that the least energy solution is trivial when $L \leq L_{*}$, the least energy solution is nontrivial when $L \in (L_{*}, L_{**}]$, and the least energy solution does not exist when $L >L_{**}$. A connection with Delaunay surfaces in CMC theory is also made.Comment: typos corrected and uniqueness adde

Predators-prey models with competition Part I: existence, bifurcation and qualitative properties

We study a mathematical model of environments populated by both preys and predators, with the possibility for predators to actively compete for the territory. For this model we study existence and uniqueness of solutions, and their asymptotic properties in time, showing that the solutions have different behavior depending on the choice of the parameters. We also construct heterogeneous stationary solutions and study the limits of strong competition and abundant resources. We then use these information to study some properties such as the existence of solutions that maximize the total population of predators. We prove that in some regimes the optimal solution for the size of the total population contains two or more groups of competing predators.Comment: 61 pages, no figure