Schr\"odinger-Poisson equations with singular potentials in R3R^3

The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schr\"{o}dinger-Poisson system {ll} -\Delta u +(\lambda+\frac{1}{|y|^\alpha})u+\phi (x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R}, -\Delta\phi = u^2,\ \lim\limits_{|x|\rightarrow +\infty}\phi(x)=0, \hfill y=(x_1,x_2) \in \mathbb{R}^2 with |y|=\sqrt{x_1^2+x_2^2}, where $\lambda\geqslant0$, $\alpha\in[0,8)$ and $\max\{2,\frac{2+\alpha}{2}\}<p<5$.Comment: 23page

Positive Eigenfunctions of a Schrödinger Operator

The paper considers the eigenvalue problem -Δu-αu+λg(x)u=0 withu∈H1(RN),u≠0 where ∞, λ ∈ and  g(x)≡0 on Ω¯,   g(x) ∈ (0,1] onRN \ Ω¯  and lim|x|→+∞g(x)=1 for some bounded open set Ω∈RN. Given α>0, does there exist a value of λ>0 for which the problem has a positive solution? It is shown that this occurs if and only if α lies in a certain interval (Γ,ξ1) and that in this case the value of λ is unique, λ=Λ(α). The properties of the function Λ(α) are also discusse

Global Branch of Solutions for Non-Linear Schrödinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation Δu+{ 1+λg(x) }u=f(u)with u∈H1RN,u ≠0 where λ > 0 and the functions f and g are such that lims→0f(s)s=0and1<α+1=lim| s |→∞f(s)s<∞d and g(x)≡0on Ω&macr;,g(x)∈(0,1]on RN\Ω&macr;and lim| x |→+∞g(x)=1 for some bounded open set Ω ∈ RN. We use topological methods to establish the existence of two connected sets D± of positive/negative solutions in R × W2, p RN where $p \in [2, \infty) \cap (\frac{N}{2},\infty)$ that cover the interval (α, Λ(α)) in the sense that PD±=(α,Λ(α))where P(λ,u)=λ, and furthermore, limλ→Λ(α)-‖ uλ ‖L∞(RN)=limλ→Λ(α)-‖ uλ ‖W2,p(RN)=∞,for (λ,uλ)∈D± The number Λ(α) is characterized as the unique value of λ in the interval (α, ∞) for which the asymptotic linearization has a positive eigenfunction. Our work uses a degree for Fredholm maps of index zero. 2000 Mathematics Subject Classification 35J60, 35B32, 58J5