Sign-changing solutions for a Schrödinger-Kirchhoff-Poisson system with 4-sublinear growth nonlinearity

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system where Ω is a bounded smooth domain of R3 , a > 0, b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q(x) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of signchanging solutions to this problem for the case that 2 < p < 4 as λ sufficiently small. Furthermore, for λ = 1 and the above assumptions on Q(x), we obtain the same conclusions with 2 < p < 12 5

Multiple and least energy sign-changing solutions for Schrodinger-Poisson equations in R3 with restraint

In this paper, we study the existence of multiple sign-changing solutions with a prescribed Lp+1−norm and theexistence of least energy sign-changing restrained solutions for the following nonlinear Schr¨odinger-Poisson system:By choosing a proper functional restricted on some appropriate subset to using a method of invariant sets of descending flow,we prove that this system has infinitely many sign-changing solutions with the prescribed Lp+1−norm and has a least energy forsuch sign-changing restrained solution for p ∈ (3, 5). Few existence results of multiple sign-changing restrained solutions areavailable in the literature. Our work generalize some results in literature

Concentration phenomena for critical fractional Schr\"odinger systems

In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schr\"odinger system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x) u=Q_{u}(u, v)+\frac{1}{2^{*}_{s}}K_{u}(u, v) &\mbox{ in } \mathbb{R}^{N}\varepsilon^{2s} (-\Delta)^{s}u+W(x) v=Q_{v}(u, v)+\frac{1}{2^{*}_{s}}K_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} u, v>0 &\mbox{ in } \R^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $W:\mathbb{R}^{N}\rightarrow \mathbb{R}$ are positive H\"older continuous potentials, $Q$ and $K$ are homogeneous $C^{2}$-functions having subcritical and critical growth respectively. We relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values. The proofs rely on the Ljusternik-Schnirelmann theory and variational methods.Comment: arXiv admin note: text overlap with arXiv:1704.0060

A uniqueness result for a Schrödinger–Poisson system with strong singularity

In this paper, we consider the following Schrödinger–Poisson system with strong singularity −∆u + φu = f(x)u , x ∈ Ω, −∆φ = u 2 , x ∈ Ω, u > 0, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where Ω ⊂ R3 is a smooth bounded domain, γ > 1, f ∈ L 1 (Ω) is a positive function (i.e. f(x) > 0 a.e. in Ω). A necessary and sufficient condition on the existence and uniqueness of positive weak solution of the system is obtained. The results supplement the main conclusions in recent literature

Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells I

We describe the asymptotic of the steady states of the out-of equilibrium Schrödinger-Poisson system, in the regime of quantum wells in a semiclassical island. After establishing uniform estimates on the nonlinearity, we show that the nonlinear steady states lie asymptotically in a finite-dimensional subspace of functions and that the involved spectral quantities are reduced to a finite number of so-called asymptotic resonant energies. The asymptotic finite dimensional nonlinear system is written in a general setting with only a partial information on its coefficients. After this first part, a complete derivation of the asymptotic nonlinear system will be done for some specific cases in a forthcoming article. UNE VERSION MODIFIEE DE CE TEXTE EST PARUE DANS LES ANNALES DE L'INSTITUT H. POINCARE, ANALYSE NON LINEAIRE