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

Supercritical problems in domains with thin toroidal holes

In this paper we study the Lane-Emden-Fowler equation $$(P)_\epsilon\ \{\Delta u+|u|^{q-2}u=0 \ \hbox{in}\ \mathcal D_\epsilon, u=0 \ \hbox{on}\ \partial\mathcal D_\epsilon.$$ Here $\mathcal D_\epsilon = \mathcal D \setminus \{x \in \mathcal D \ : \ \mathrm{dist}(x,\Gamma_\ell)\le \epsilon\}$, $\mathcal D$ is a smooth bounded domain in $\mathbb{R}^N$, $\Gamma_\ell$ is an $\ell-$dimensional closed manifold such that $\Gamma_\ell \subset \mathcal D$ with $1\le \ell \le N-3$ and $q={2(N-\ell)\over N-\ell-2}$. We prove that, under some symmetry assumptions, the number of sign changing solutions to $(P)_\epsilon$ increases as $\epsilon$ goes to zero