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

Symmetries, Hopf fibrations and supercritical elliptic problems

We consider the semilinear elliptic boundary value problem $$-\Delta u=\left\vert u\right\vert ^{p-2}u\text{ in }\Omega,\text{\quad }u=0\text{ on }\partial\Omega,$$ in a bounded smooth domain $\Omega$ of $\mathbb{R}^{N}$ for supercritical exponents $p>\frac{2N}{N-2}.$ Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents

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

Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem -\Delta u = |u|^{p-2}u in \Omega, u=0 on \partial\Omega, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $\Omega$ has nontrivial homology this problem has a positive solution for $p=2^{*}.$ However, this is not enough to guarantee existence in the supercritical case. For $p\geq 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show that for $p> 2(N-k)/(N-k-2)$ with $1\leq k\leq N-3$ there are bounded smooth domains in $\mathbb{R}^{N}$ whose cup-length is $k+1$ in which this problem does not have a nontrivial solution. For $N=4,8,16$ we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.Comment: Published online in Calculus of Variations and Partial Differential Equation