Generalized Schr\"odinger-Newton system in dimension N≥3N\ge 3: critical case

In this paper we study a system which is equivalent to a nonlocal version of the well known Brezis Nirenberg problem. The difficulties related with the lack of compactness are here emphasized by the nonlocal nature of the critical nonlinear term. We prove existence and nonexistence results of positive solutions when $N=3$ and existence of solutions in both the resonance and the nonresonance case for higher dimensions.Comment: 18 pages, typos fixed, some minor revision

Infinitely many positive solutions for a Schrodinger-Poisson system

We find infinitely many positive non-radial solutions for a nonlinear Schrodinger-Poisson system.Comment: 23 page

Bubbling solutions for supercritical problems on manifolds

Let $(M,g)$ be a $n-$dimensional compact Riemannian manifold without boundary and $\Gamma$ be a non degenerate closed geodesic of $(M,g)$. We prove that the supercritical problem $$-\Delta_gu+h u=u^{\frac{n+1}{n-3}\pm\epsilon},\ u>0,\ \hbox{in}\ (M,g)$$ has a solution that concentrates along $\Gamma$ as $\epsilon$ goes to zero, provided the function $h$ and the sectional curvatures along $\Gamma$ satisfy a suitable condition. A connection with the solution of a class of periodic O.D.E.'s with singularity of attractive or repulsive type is established

Large mass boundary condensation patterns in the stationary Keller-Segel system

We consider the boundary value problem $-\Delta u + u =\lambda e^u$ in $\Omega$ with Neumann boundary condition, where $\Omega$ is a bounded smooth domain in $\mathbb R^2$, $\lambda>0.$ This problem is equivalent to the stationary Keller-Segel system from chemotaxis. We establish the existence of a solution $u_\lambda$ which exhibits a sharp boundary layer along the entire boundary $\partial\Omega$ as $\lambda\to 0$. These solutions have large mass in the sense that $ \int_\Omega \lambda e^{u_\lambda} \sim |\log\lambda|.