Ricerc@Sapienza

We consider the following critical weakly coupled elliptic system -Δui=μi|ui|2-2ui+j≠iβij|uj|22|ui|2-42uiinΩϵui=0on∂Ωϵ,i=1,m,in a domain Ωϵ⊂ℝN, N=3,4, with small shrinking holes as the parameter ϵ→0. We prove the existence of positive solutions of two different types: either each density concentrates around a different hole, or we have groups of components such that all the components within a single group concentrate around the same point, and different groups concentrate around different points.

We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator and a concave-convex term, together with mixed Dirichlet-Neumann boundary conditions.

We analyze the behavior of the eigenvalues of the following non local mixed
problem $\left\ \beginarrayrcll (-\Delta)^s u &=& \lambda_1(D) \ u
&\inn\Omega,\\ u&=&0&\inn D,\\ \mathcalN_su&=&0&\inn N. \endarray\right $
Our goal is to construct different sequences of problems by modifying the
configuration of the sets $D$ and $N$, and to provide sufficient and necessary
conditions on the size and the location of these sets in order to obtain
sequences of eigenvalues that in the limit recover the eigenvalues of the