Existence and phase separation of entire solutions to a pure critical competitive elliptic system

We establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \left\{ \begin{tabular} [c]{l}% $-\Delta u=\mu_{1}|u|^{{2}^{\ast}-2}u+\lambda\alpha|u|^{\alpha-2}|v|^{\beta }u,$\\ $-\Delta v=\mu_{2}|v|^{{2}^{\ast}-2}v+\lambda\beta|u|^{\alpha}|v|^{\beta{-2}% }v,$\\ $u,v\in D^{1,2}(\mathbb{R}^{N}),$% \end{tabular} \ \right. where $N\geq4,$ $2^{\ast}:=\frac{2N}{N-2}$ is the critical Sobolev exponent, $\alpha,\beta\in(1,2],$ $\alpha+\beta=2^{\ast},$ $\mu_{1},\mu_{2}>0,$ and $\lambda<0.$ We show that these solutions exhibit phase separation as $\lambda\rightarrow-\infty,$ and we give a precise description of their limit domains. If $\mu_{1}=\mu_{2}$ and $\alpha=\beta$, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent

Spiked solutions for Schr\"odinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions

In this paper we deal with the nonlinear Schr\"odinger system $$-\Delta u_i =\mu_i u_i^3 + \beta u_i \sum_{j\neq i} u_j^2 + \lambda_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(\Omega)$$ in dimension 4, a problem with critical Sobolev exponent. In the competitive case ($\beta<0$ fixed or $\beta\to -\infty$) or in the weakly cooperative case ($\beta\geq 0$ small), we construct, under suitable assumptions on the Robin function associated to the domain $\Omega$, families of positive solutions which blowup and concentrate at different points as $\lambda_1,\ldots, \lambda_m\to 0$. This problem can be seen as a generalization for systems of a Brezis-Nirenberg type problem.Comment: 33 page

Boundary towers of layers for some supercritical problems

We show that in some suitable torus-like domains D some supercritical elliptic problems have an arbitrary large number of sign-changing solutions with alternate positive and negative layers which concentrate at different rates along a k-dimensional submanifold of the boundary of D as p approaches 2*_{N,K} from below