Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation

Abstract

<jats:title>Abstract</jats:title><jats:p>In this paper, we study the following nonlinear magnetic Schrödinger-Poisson type equation</jats:p><jats:p><jats:disp-formula id="j_anona-2020-0110_eq_001_w2aab3b7d967b1b6b1aab1c15b2aAa"><jats:alternatives><jats:tex-math>$$\begin{array}{} \displaystyle \left\{\!\begin{aligned}&amp;\Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u = f(|u|^{2})u\quad\hbox{in }\mathbb{R}^3,\\&amp;u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}),\end{aligned}\right. \end{array}$$</jats:tex-math></jats:alternatives></jats:disp-formula></jats:p><jats:p>where <jats:italic>ϵ</jats:italic> &gt; 0, <jats:italic>V</jats:italic> : ℝ<jats:sup>3</jats:sup> → ℝ and <jats:italic>A</jats:italic> : ℝ<jats:sup>3</jats:sup> → ℝ<jats:sup>3</jats:sup> are continuous potentials. Under a local assumption on the potential <jats:italic>V</jats:italic>, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration properties of nontrivial solutions for <jats:italic>ε</jats:italic> &gt; 0 small. In this problem, the function <jats:italic>f</jats:italic> is only continuous, which allow to consider larger classes of nonlinearities in the reaction.</jats:p>

Concentration results for a magnetic Schrödinger-Poisson system with critical growth

“…If f is only continuous, then the arguments in [41] failed. Recently, by variational methods, penalization technique, and Ljusternick-Schniremann theory, for the magnetic Schrödinger-Poisson system with subcritical growth nonlinearity f which is only continuous, in [29] we proved multiplicity and concentration properties of nontrivial solutions for ϵ > small. For the fractional Schrödinger-Poisson type equations with magnetic eld, we refer to [2,3].…”

“…Inspried by [27,29], we intend to prove multiplicity and concentration of nontrivial solutions for problem (1.1) with critical growth. Since the probem we deal with has the critical growth, we need more re ned estimates to overcome the lack of compactness.…”

“…Proof. (A ) Arguing as in [29,Lemma 3.1], it follows that gu( ) = , gu(t) > for t > small and gu(t) < for t > large. Thus, max t≥ gu(t) is achieved at a global maximum point t = tu satisfying g u (tu) = and tu u ∈ Nϵ.…”

Linear Barycentric Rational Method for Solving Schrodinger Equation

“…In [2], the finite-difference time-domain (FDTD) method is studied to solve SDE. In [3], nonlinear magnetic Schrodinger-Poisson type equation is studied. In [4], high-order multiscale discontinuous Galerkin method for one-dimensional stationary SDEs with oscillating solutions is presented.…”

Ground‐state solution of a nonlinear fractional Schrödinger–Poisson system

“…Here, 𝜀 > 0 is a small parameter. Existence, multiplicity, and asymptotic behavior as 𝜀 → 0 + of semiclassical solutions of (1.3) are proved in several studies [23][24][25][26][27][28] via variational methods or reduction methods.…”