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We consider $L^{2}$ -constraint minimizers of the mass critical fractional Schrödinger energy functional with a ring-shaped potential $V(x)=(|x|-M)^{2}$ , where $M>0$ and $x\in \mathbb {R}^{2}$. By analysing some new estimates on the least energy of the mass critical fractional Schrödinger energy functional, we obtain the concentration behaviour of each minimizer of the mass critical fractional Schrödinger energy functional when $a\nearrow a^{\ast }=\|Q\|_{2}^{2s}$ , where $Q$ is the unique positive radial solution of $(-\Delta)^{s}u+su-|u|^{2s}u=0$ in $\mathbb {R}^{2}$. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of critical Schrödinger-Bopp-Podolsky system. By virtue of the Nehari manifold and variational methods, we study the existence, nonexistence and asymptotic behavior of ground state solutions for this problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Schrödinger–Poisson equations: { - ε 2 ⁢ Δ ⁢ u + V ⁢ (x) ⁢ u + K ⁢ (x) ⁢ ϕ ⁢ u = | u | p - 2 ⁢ u , x ∈ ℝ 3 , - ε 2 ⁢ Δ ⁢ ϕ = K ⁢ (x) ⁢ u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\varepsilon^{2}\Delta u+V(x)u+K(x)\phi u% =\lvert u\rvert^{p-2}u,&\hskip 10.0ptx&\displaystyle\in\mathbb{R}^{3},\\ &\displaystyle{-}\varepsilon^{2}\Delta\phi=K(x)u^{2},&\hskip 10.0ptx&% \displaystyle\in\mathbb{R}^{3},\end{aligned}\right. where p ∈ (4 , 6) {p\in(4,6)} , ε > 0 {\varepsilon>0} is a parameter, and V and K are nonnegative potential functions which satisfy the critical frequency conditions in the sense that inf ℝ 3 ⁡ V = inf ℝ 3 ⁡ K = 0 {\inf_{\mathbb{R}^{3}}V=\inf_{\mathbb{R}^{3}}K=0}. By using a penalization method, we show the existence of multi-bump solutions for the above problem, with several local maximum points whose corresponding values are of different scales with respect to ε → 0 {\varepsilon\rightarrow 0}. Moreover, under suitable local assumptions on V and K, we prove the uniqueness of multi-bump solutions concentrating around zero points of V and K via the local Pohozaev identity. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following two-component system of Schrödinger equations with Hartree nonlinearity:where is a small parameter, , , and are constants. Under some suitable assumptions on the potentials and the coupled function , we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic equation: BvΔ2u+Vλ(x)u=α(x)f(u)+νK(x)|u|q−2uinRN,u∈H2(RN),where N⩾5,ν ∈ (0,ν0],1  <  q  <  2,Δ2u  =  Δ(Δu) and Vλ(x)  =  λa(x) − b(x) with λ > 0. Firstly, we prove the bipolar Rellich inequality. Secondly, by using bipolar Rellich inequality, Gigliardo‐Nirenberg inequality, and Ekeland variational principle, we prove the existence of nontrivial solution for problem Bν. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the following quasilinear Schrödinger equations: { − △ u + V (x) u − △ (u 2) u = λ u p − 1 u in R N , u > 0 , u ∈ H 1 (R N) , where λ > 0 , 1 < p < 22 ⁎ − 1 , 2 ⁎ = 2 N N − 2 and N ≥ 3. Under some suitable assumptions on V (x) , we prove the existence of ground state solutions via Pohožaev manifold method. The novelty of this works with respect to some recent results is that we treat the existence by using Pohožaev manifold method in an Orlicz space and this enables us to handle the nonlinearity in a uniform way. As its supplementary results, we also prove the nonexistence of least energy solutions for 1 < p < 22 ⁎ − 1. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following fractional Schrödinger‐Poisson system ( − Δ ) α u + V ( x ) u + K ( x ) ϕ u = f ( x ) | u | q − 2 u , in R 3 , ( − Δ ) t ϕ = K ( x ) u 2 , in R 3 , where 0 < t ≤ α < 1, q ∈ ( 4 , 2 α ∗ ), and 4α+2t ≥3 and the functions V(x), K(x) and f(x) have finite limits as |x|→∞. By imposing some suitable conditions on the decay rate of the functions, we prove that the above system has two nontrivial solutions. One of them is positive and the other one is sign‐changing. Recent results from the literature are generally improved and extended. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Schrödinger equation − Δ u + V λ ( x ) u + μ ϕ | u | p − 2 u = f ( x , u ) + β ( x ) | u | ν − 2 u , in R 3 , ( − Δ ) α 2 ϕ = μ | u | p , in R 3 , where μ ≥ 0 is a parameter, α ∈ ( 0 , 3 ) , ν ∈ ( 1 , 2 ) and p ∈ [ 2 , 3 + 2 α ) . V λ is allowed to be sign-changing and ϕ | u | p − 2 u is a Hartree-type nonlinearity. We require that V λ = λ V + − V − with V + having a bounded potential well Ω whose depth is controlled by λ . Under some mild conditions on V λ ( x ) and f ( x , u ) , we prove that the above system has at least two nontrivial solutions. Specially, our results cover the general Schrödinger equations and the Schrödinger–Poisson equations. [ABSTRACT FROM AUTHOR]

This paper focuses on the following generalized quasilinear Schrödinger equations: -div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=f(x,u),x∈RN,where N≥3 and g:R→R+ is an even differential function. By using the methods of perturbation and the Mountain Pass Theorem, we obtain the existence and multiplicity of bound state solutions for this problem with a finite depth potential well, extending the recent results from the literature. [ABSTRACT FROM AUTHOR]

We study the existence of ground state solutions for the following Schrödinger-Poisson equations: [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger equations: [ABSTRACT FROM AUTHOR]

This paper is concerned with the concentration of positive ground states solutions for a modified Schrödinger equation where $$40$$ is a parameter and $$2^{*}:=\frac{2N}{N-2}(N\ge 3)$$ is the critical Sobolev exponent. We prove the existence of a positive ground state solution $$v_{\varepsilon }$$ and $$\varepsilon $$ sufficiently small under some suitable conditions on the nonnegative functions V( x) and K( x). Moreover, $$v_{\varepsilon }$$ concentrates around a global minimum point of V as $$\varepsilon \rightarrow 0$$ . The proof of the main result is based on minimax theorems and concentration compact theory. [ABSTRACT FROM AUTHOR]

This paper focuses on the following generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = f ( x , u ) , x ∈ R N , where N ≥ 3 , g ( s ) : R → R + is a nondecreasing function with respect to | s | . By using a change of variables and variational methods, we obtain the existence and multiplicity of nontrivial solutions for the above problem when the nonlinearity is superlinear but does not satisfy the Ambrosetti–Rabinowitz type condition. [ABSTRACT FROM AUTHOR]

This paper deals with the following Schrödinger-Poisson systems [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasilinear Schrödinger equation: where $N\geq3$ , V is a nonnegative continuous function, which can vanish at infinity, and g is a continuous function with a quasicritical growth. Under some appropriate conditions, we get the existence of a positive solution by combining the variational method with a Hardy-type inequality. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following Schrödinger-Kirchhoff-Poisson system: where $a\geq0$ , $b>0$ and $\eta,\lambda>0$ , $\Omega\subset R^{3}$ is a bounded smooth domain. With the help of the variational methods, the existence of a non-trivial solution is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Schrödinger equations ( − Δ ) s u + V ( x ) u = f ( x , u ) , x ∈ R N , where s ∈ ( 0 , 1 ) , N > 2 s , ( − Δ ) s stands for the fractional Laplacian. Under more relaxed assumption on f ( x , u ) , we obtain a new existence result of infinitely many high energy solutions via Symmetric Mountain Pass Theorem, which unifies and improves Theorem 1.2. in Teng (2015). [ABSTRACT FROM AUTHOR]

This paper is concerned with the following quasilinear Schrödinger equations: { − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) ∣ ∇ u ∣ 2 + V ( x ) u = K ( x ) f ( u ) , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where N ≥ 3 and V , K are nonnegative continuous functions. Firstly by using a change of variables, the quasilinear equation is reduced to a semilinear one, whose associated functional is still not well defined in D 1 , 2 ( R N ) because of the potential vanishing at infinity. However, by using a Hardy-type inequality, we can work in the weighted Sobolev space in which the functional is well defined. Using this fact together with the variational methods, we obtain a positive solution. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two kinds of Lie conformal algebras, associated with the loop Schrödinger-Virasoro Lie algebra and the extended loop Schrödinger-Virasoro Lie algebra, respectively. The conformal derivations, the second cohomology groups of these two conformal algebras are completely determined. And nontrivial free conformal modules of rank one and -graded free intermediate series modules over these two conformal algebras are also classified in the present paper. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Schrödinger-Kirchhoff-type equations: where $a, b,\mu>0$ are constants, $2^{*}=6$ is the critical Sobolev exponent in three spatial dimensions. Under appropriate assumptions on nonnegative functions $k(x)$ and $h(x)$, we establish the existence of positive and sign-changing solutions by variational methods. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of nontrivial solutions for a nonlinear Schrödinger–Poisson system. Under some suitable conditions, some criteria on the existence of nontrivial solutions, including a ground state solution, two solutions and a sequence of nontrivial solutions { u k } with ‖ u k ‖ → 0 as k → + ∞ , is established by applying the Morse theory and variational methods. Some known results in the literatures are extended and improved. [ABSTRACT FROM AUTHOR]

We consider the following nonlinear stationary Schrödinger equation − Δ u + λ V ( x ) u = K ( x ) f ( u ) , in R N , where N ≥ 3 , λ > 0 and V ( x ) changes sign and may vanish at infinity. Under some suitable conditions, the least energy solution is obtained by using variational methods. Moreover, the solution changes sign when λ is sufficiently large. Our result unifies and improves the recent one in the literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following quasilinear Schrödinger equations: { − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = λ f ( x , u ) + g ( u ) | G ( u ) | 2 ∗ − 2 G ( u ) , x ∈ R N , u ( x ) > 0 , x ∈ R N , where N ≥ 3 , 2 ∗ = 2 N N − 2 , G ( u ) = ∫ 0 u g ( t ) d t and λ > 0 is a parameter. By using a change of variable, the quasilinear equation is reduced to a semilinear one, whose associated functional is well defined in H 1 ( R N ) . We establish the existence of positive solutions for this problem by using the Mountain Pass Theorem in combination with the concentration-compactness principle under appropriate assumptions on V ( x ) and f ( x , u ) . Recent results from the literature are improved and extended. [ABSTRACT FROM AUTHOR]

In this paper, we study a 4-sublinear Schrödinger–Poisson system with sign-changing potential. Under some suitable assumptions, the existence of two nontrivial solutions are obtained by using the Morse theory. Our result improves the recent ones of Chen and Zhang (2014) [6] . [ABSTRACT FROM AUTHOR]

In this paper, we study the following Schrödinger–Poisson system { − Δ u + V ( x ) u + λ ϕ ( x ) u = f ( x , u ) , in R 3 , − Δ ϕ = u 2 , in R 3 , where the nonlinearity f and the potential V are allowed to be sign-changing. Under some appropriate assumptions on V and f , we obtain the existence of two different solutions of the system via the Ekeland’s variational principle and the Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the Schrödinger–Poisson system and prove the existence of ground state solutions for system (SP) under certain assumptions on the linear and nonlinear terms. Some recent results from different authors are extended. [Copyright &y& Elsevier]

Abstract: In this paper, we study the Schrödinger–Poisson system and prove the existence of positive solutions for system (SP) when the nonlinearity has growth at most linear for small, allowing the potential to vanish at infinity. In addition, also we obtain the nonexistence of a nontrivial positive solution for . [Copyright &y& Elsevier]