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We establish the existence of positive normalized (in the $L^2$ sense) solutions to non-variational weakly coupled elliptic systems of $\ell$ equations. We consider couplings of both cooperative and competitive type. We show the problem can be formulated as an operator equation on the product of $\ell$ $L^2$-spheres and apply a degree-theoretical argument on this product to obtain existence.

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.

We consider the problem \[ -\Delta u=|u|^{p-2}u in \Omega, u=0 on \partial\Omega, \] where $\Omega:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}: 0

We consider the semilinear electromagnetic Schr��dinger equation (-i\nabla+A(x))^{2}u + V(x)u = |u|^{2^{\ast}-2}u, u\in D_{A,0}^{1,2}(��,\mathbb{C}), where $��=(\mathbb{R}^{m}\smallsetminus{0})\times\mathbb{R}^{N-m}$ with $2\leq m\leq N$, $N\geq3$, $2^{\ast}:= 2N/(N-2)$ is the critical Sobolev exponent, $V$ is a Hardy term and $A$ is a singular magnetic potential of a particular form which includes the Aharonov-Bohm potentials. Under some symmetry assumptions on $A$ we obtain multiplicity of solutions satisfying certain symmetry properties., Journal of Fixed Point Theory and Applications, to appear

We study the problem (-��\mathrm{i}\nabla+A(x)) ^{2}u+V(x)u=��^{-2}(\frac{1}{|x|}\ast|u|^{2}) u, u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),\text{\ \ \ \}��\nabla u+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}), where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magnetic potential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exterior electric potential, and $��$ is a small positive number. If A=0 and $��=\hbar$ is Planck's constant this problem is equivalent to the Schr��dinger-Newton equations proposed by Penrose in \cite{pe2}\ to describe his view that quantum state reduction occurs due to some gravitational effect. We assume that $A$ and $V$ are compatible with the action of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for any given homomorphism $��:G\rightarrow\mathbb{S}^{1}$ into the unit complex numbers, we show that there is a combined effect of the symmetries and the potential $V$ on the number of semiclassical solutions $u:\mathbb{R}% ^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=��(g)u(x)$ for all $g\in G$, $x\in\mathbb{R}^{3}$. We also study the concentration behavior of these solutions as $��\rightarrow0.\medskip$, 18 pages, to appear on DCDS-S