Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in {\mathbb{R}^{3}}

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{array}\right.$$ where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that a(x) has at least one minimum and b(x) has at least one maximum. We first prove the existence of least energy solution (u _ε , φ _ε ) for λ ≠ 0 and ε > 0 sufficiently small. Then we show that u _ε converges to the least energy solution of the associated limit problem and concentrates to some set. At the same time, some properties for the least energy solution are also considered. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.     

On the Fučik point spectrum for Schrödinger operators on {\mathbb{R}}^{N}

We investigate the Fučik point spectrum of the Schr?dinger operator when the potential V_λ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that S_λ has essential spectrum with finitely many eigenvalues below the infimum of . We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schr?dinger equations with jumping nonlinearity.      

Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in {\mathbb{R}^N}

We consider the following perturbed version of quasilinear Schrödinger equation $$\begin{array}{lll}-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=h(x,u)u+K(x)|u|^{22^*-2}u\end{array}$$ in ${\mathbb{R}^N}$ , where N ≥ 3, 22* = 4N/(N ? 2), V(x) is a nonnegative potential, and K(x) is a bounded positive function. Using minimax methods, we show that this equation has at least one positive solution provided that ${\varepsilon \leq \mathcal{E}}$ ; for any ${m\in\mathbb{N}}$ , it has m pairs of solutions if ${\varepsilon \leq \mathcal{E}_m}$ , where ${\mathcal{E}}$ and ${\mathcal{E}_m}$ are sufficiently small positive numbers. Moreover, these solutions ${u_\varepsilon \to 0}$ in ${H^1(\mathbb{R}^N)}$ as ${\varepsilon \to 0}$ .     

Existence of Positive Solutions for a Class of Critical Fractional Schrödinger Equations with Potential Vanishing at Infinity

In this paper, we investigate the following critical fractional Schrödinger equation

$$\begin{aligned} (-\Delta )^su+V(x)u=|u|^{2_s^*-2}u+\lambda K(x)f(u), \ x \in \mathbb {R}^N, \end{aligned}$$

where \(\lambda >0\), \(0<s<1\), \((-\Delta )^s\) denotes the fractional Laplacian of order s, \(V, \ K\) are nonnegative continuous functions satisfying some conditions and f is a continuous function, \(N>2s\) and \(2_s^*=\frac{2N}{N-2s}\). We prove that the equation has a positive solution for large \(\lambda \) by the variational method.

     

A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $${\mathbb R^{N}}$$

In this work we study the following class of problems in \({\mathbb R^{N}, N > 2s}\)

$$\varepsilon^{2s}(-\Delta)^{s}u + V(z)u = f(u), \,\,\,u(z) > 0$$

where \({0 < s < 1}\), \({(-\Delta)^{s}}\) is the fractional Laplacian, \({\varepsilon}\) is a positive parameter, the potential \({V : \mathbb{R}^N \to \mathbb{R}}\) and the nonlinearity \({f : \mathbb R \to \mathbb R}\) satisfy suitable assumptions; in particular it is assumed that \({V}\) achieves its positive minimum on some set \({M.}\) By using variational methods we prove existence and multiplicity of positive solutions when \({\varepsilon \to 0^{+}}\). In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set \({M}\).

     

On the Schröder equation and iterative sequences of C r diffeomorphisms in {\mathbb{R}^{N}} space

Let ${U \subset \mathbb{R}^{N}}$ be a neighbourhood of the origin and a function ${F:U\rightarrow U}$ be of class C ^r , r ≥ 2, F(0) = 0. Denote by F ⁿ the n-th iterate of F and let ${0<|s_1|\leq \cdots \leq|s_N| <1 }$ , where ${s_1, \ldots , s_N}$ are the eigenvalues of dF(0). Assume that the Schröder equation ${\varphi(F(x))=S\varphi(x)}$ , where S: = dF(0) has a C ² solution φ such that dφ(0) = id. If ${\frac{log|s_1|}{log|s_N|} <2 }$ then the sequence {S ^?n F ⁿ (x)} converges for every point x from the basin of attraction of F to a C ² solution φ of (1). If ${2\leq\frac{log|s_1|}{log|s_N|} }$ then this sequence can be diverging. In this case we give some sufficient conditions for the convergence and divergence of the sequence {S ^?n F ⁿ (x)}. Moreover, we show that if F is of class C ^r and ${r>\big[\frac{log|s_1|}{log|s_N|} \big ]:=p \geq 2}$ then every C ^r solution of the Schröder equation such that dφ(0) = id is given by the formula $$\begin{array}{ll}\varphi (x)={\lim\limits_{n \rightarrow \infty}} (S^{-n}F^n(x) + {\sum\limits _{k=2}^{p}} S^{-n}L_k (F^n(x))),\end{array}$$ where ${L_k:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}}$ are some homogeneous polynomials of degree k, which are determined by the differentials d ^(j) F(0) for 1 < j ≤  p.     

Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

Acta Mathematicae Applicatae Sinica, English Series - We study the following quasilinear Schrödinger equation $$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in...     

Infinitely many positive solutions for the nonlinear Schrödinger equations in {\mathbb{R}^N}

We consider the following nonlinear problem in ${\mathbb {R}^N}$ $$- \Delta u +V(|y|)u = u^{p},\quad u > 0 \quad {\rm in}\, \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \quad \quad \quad (0.1)$$ where V(r) is a positive function, ${1< p < {\frac{N+2}{N-2}}}$ . We show that if V(r) has the following expansion: $$V(r) = V_0+\frac a {r^m} +O \left(\frac1{r^{m+\theta}}\right),\quad {\rm as} \, r\to +\infty,$$ where a > 0, m > 1, θ > 0, and V ₀ > 0 are some constants, then (0.1) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.     

Multi-bump Bound States of Schrödinger Equations with a Critical Frequency

We consider existence and qualitative properties of standing wave solutions $\Psi(x,t) = e^{-iEt/h}u(x)We consider existence and qualitative properties of standing wave solutions to the nonlinear Schr?dinger equation with E being a critical frequency in the sense that inf . We verify that if the zero set of W − E has several isolated points x _i () near which W − E is almost exponentially flat with approximately the same behavior, then for h > 0 small enough, there exists, for any integer k, , a standing wave solution which concentrates simultaneously on , where is any given subset of . This generalizes the result of Byeon and Wang in 3 (Arch Rat Mech Anal 165: 295–316, 2002).Supported by the Alexander von Humboldt foundation and NSFC(No:10571069).     

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).     

Solutions for Quasilinear Schrödinger Equations via the Nehari Method

Abstract

For a class of quasilinear Schrödinger equations we establish the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method.     

On a Schrödinger equation with periodic potential and critical growth in \mathbb{R}^{2}

The main purpose of this paper is to establish the existence of a solution of the semilinear Schr?dinger equation