Search

We consider the fractional Schrödinger-Poisson equation \(\begin{cases}(-\Delta)^su-\lambda u+\phi u=|u|^{p-2}u,& x\in\mathbb{R}^3,\\ (-\Delta)^t\phi=u^2,& x\in\mathbb{R}^3,\end{cases}\) where \(s,t\in(0,1)\) satisfies \(2s+2t>3\), \(p\in(\frac{4s+6}{3},2^*_s)\) and \(\lambda\in\mathbb{R}\) is an undetermined parameter. We deal with the case where the associated functional is not bounded below on the \(L^2\)-unit sphere and show the existence of infinitely many solutions \((u,\lambda)\) with \(u\) having prescribed \(L^2\)-norm.

In this paper, we study the singularly perturbed fractional Choquard equationε2s(−Δ)su+V(x)u=εμ−3(∫R3|u(y)|2μ,s∗+F(u(y))|x−y|μdy)(|u|2μ,s∗−2u+12μ,s∗f(u))inR3,$$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$whereε> 0 is a small parameter, (−△)sdenotes the fractional Laplacian of orders ∈(0, 1), 0 <μ< 3,2μ,s⋆=6−μ3−2s$2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator.Fis the primitive offwhich is a continuous subcritical term. Under a local condition imposed on the potentialV, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

In this article, we are concerned with the existence of solutions of a quasilinear elliptic equation in ℝN which includes the so-called modified nonlinear Schrodinger equation as a special case. Combining the dual approach and the nonsmooth critical point theory, we obtain the existence of a nontrivial solution.

In this paper, we study the following fractional Schrodinger–Poisson system (FSP) ( − Δ ) s u + u + K ( x ) φ u = h ( x ) | u | p − 2 u + | u | 2 s ∗ − 2 u , in R 3 , ( − Δ ) t φ = K ( x ) u 2 , ...

In this paper, we study the existence and multiplicity of solutions for the following fractional Schrodinger-Poisson system: (0.1) $$\left\{ \begin{gathered} {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}u + V\left( x \right)u + \phi u = {\left| u \right|^{2_s^* - 2}}u + f\left( u \right)in{\mathbb{R}^3}, \hfill \\ {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}\phi = {u^2}in{\mathbb{R}^3}, \hfill \\ \end{gathered} \right.$$ where $$\frac{3}{4} < s < 1, 2_s^*:=\frac{6}{3-2s}$$ the fractional critical exponent for 3-dimension, the potential V: ℝ3 → ℝ is continuous and has global minima, and f is continuous and supercubic but subcritical at infinity. We prove the existence and multiplicity of solutions for System (0.1) via variational methods.

In this paper, we consider the following general nonlocal problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\mathcal {L}_K u=f(x,u)&{}\text {in} \,\,\varOmega ,\\ u=0&{}\text {in}\,\,\mathbb {R}^N\backslash \varOmega , \end{array} \right. \end{aligned}$$ where $$\varOmega \subset \mathbb {R}^N$$ is a bounded domain with Lipschitz boundary $$\partial \varOmega $$ , $$s\in (0,1)$$ with $$2s

EG27I is an endogenous glucanase belonging to glycoside hydrolase family (GHF) 45 from the mollusk Ampullaria crossean. In this study, the mature EG27I peptide gene fused to the HFBII secretion signal of Trichoderma reesei was expressed under the GAP promoter of Pichia pastoris in SMD1163 strain. A bioactive EG27I with a molecular weight of 27 kDa was successfully expressed and secreted into our culture medium. The respective final OD600 and hydrolytic activity were 333 and 1.28 U/mL when high-cell-density fermentation of the recombinant P. pastoris was performed in a 7.5 L fermenter through a fed-batch strategy for 132 h. The recombinant protein concentration of the fermentation supernatant was 47.7 mg/L. EG27I was consecutively purified from the fermentation supernatant through ultrafiltration, cation exchange, and hydrophobic interaction. The specific activity of the recombinant EG27I was 26.8 U/mg, and the optimal pH and temperature of the enzyme were 5 and 50 °C, respectively. The half-life of the enzyme activity at 100 °C could reach 40 min. The N-terminal amino acid sequence analysis of the purified recombinant protein confirmed that the amino terminal sequence was consistent with the natural structure. The high quantity and purity of the EG27I provide a basis for future structural and functional studies.

We consider the following nonlocal critical equation \begin{document}$\begin{equation} -\Delta u = (I_{\mu_1}\ast|u|^{2_{\mu_1}^\ast})|u|^{2_{\mu_1}^\ast-2}u +(I_{\mu_2}\ast|u|^{2_{\mu_2}^\ast})|u|^{2_{\mu_2}^\ast-2}u,\; x\in\mathbb{R}^N, \;\;\;\;\;\;\;(1) \end{equation}$ \end{document} where \begin{document}$ 0 if \begin{document}$ N = 3 $\end{document} or \begin{document}$ 4 $\end{document}, and \begin{document}$ N-4\leq\mu_1,\mu_2 if \begin{document}$ N\geq5 $\end{document}, \begin{document}$ 2_{\mu_{i}}^\ast: = \frac{N+\mu_i}{N-2}(i = 1,2) $\end{document} is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and \begin{document}$ I_{\mu_i} $\end{document} is the Riesz potential \begin{document}$ \begin{equation*} I_{\mu_i}(x) = \frac{\Gamma(\frac{N-\mu_i}{2})}{\Gamma(\frac{\mu_i}{2})\pi^{\frac{N}{2}}2^{\mu_i}|x|^{N-\mu_i}}, \; i = 1,2, \end{equation*} $\end{document} with \begin{document}$ \Gamma(s) = \int_{0}^{\infty}x^{s-1}e^{-x}dx $\end{document}, \begin{document}$ s>0 $\end{document}. Firstly, we prove the existence of the solutions of the equation (1). We also establish integrability and \begin{document}$ C^\infty $\end{document}-regularity of solutions and obtain the explicit forms of positive solutions via the method of moving spheres in integral forms. Finally, we show that the nondegeneracy of the linearized equation of (1) at \begin{document}$ U_0,V_0 $\end{document} when \begin{document}$ \max\{\mu_1,\mu_2\}\rightarrow0 $\end{document} and \begin{document}$ \min\{\mu_1,\mu_2\}\rightarrow N $\end{document}, respectively.

In this paper, we study the following nonlinear Dirac equation $$\begin{aligned} -i\sum _{k=1}^{3}\alpha _{k}\partial _{k}u+a\beta u+V(x)u=f(x,|u|)u,~~x\in \mathbb {R}^{3}. \end{aligned}$$ The Dirac operator is unbounded from below and above so the associate energy functional is strongly indefinite. Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground-state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground-state solutions, such as compactness of set of ground-state solutions and exponential decay of ground-state solutions.

This paper is concerned with the following nonperiodic Hamiltonian elliptic system { - Δ u + V ( x ) u = H v ( x , u , v ) x ? R N , - Δ v + V ( x ) v = H u ( x , u , v ) x ? R N , u ( x ) ? 0 , v ( x ) ? 0 as | x | ? ∞ , where z = ( u , v ) : R N ? R i? R , N ? 3, and the potential V(x) is nonperiodic and sign-changing. By applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of multiple solutions for asymptotically quadratic nonlinearity as well as the existence of infinitely many solutions for superquadratic nonlinearity.

Schrodinger type soliton waves generated by second-harmonic generation in higher dimensional quadratic optical media are considered. The existence of ground state solutions for spatial dimension from two to five is proved, and the continuous dependence on the parameter and asymptotic behavior of ground state solutions are established. Multi-pulse solutions with certain symmetry are also obtained. In a bounded domain setting, global bifurcation diagram of multi-pulse solutions are shown by using new technique of double saddle-node bifurcation.

In this paper, we study the following quasilinear Schrodinger equation − e 2 Δ u + V ( x ) u − e 2 1 2 Δ ( u 2 ) u = K ( x ) | u | p − 1 u , where 1 p 3 , e > 0 is a small parameter, and V ( x ) has global minimum and K ( x ) has global maximum. We show the existence of ground state for e > 0 via a constrained minimization on Pohozaev manifold. Furthermore, we prove these ground state solutions concentrate at some set related to the linear potential V ( x ) and the nonlinear potential K ( x ) .

In this paper, we are concerned with the following quasilinear Schrodinger equation − Δ u − u Δ ( ∣ u ∣ 2 ) + V ( ∣ x ∣ ) u = f ( ∣ x ∣ , u ) , x ∈ R N . By using a change of variables, we obtained the existence of a sequence of radial solutions for N ≥ 2 , a sequence of nonradial solutions for N = 4 or N ≥ 6 , and a nonradial solution for N = 5 .

In this paper, we study the following fractional Schrodinger–Poisson system 0.1 $$\begin{aligned} \left\{ \begin{array}{ll} \varepsilon ^{2s}(-\Delta )^s u +V(x)u+\phi u=K(x)|u|^{p-2}u,\,\,\text {in}~\mathbb {R}^3,\\ \\ \varepsilon ^{2s}(-\Delta )^s \phi =u^2,\,\,\text {in}~\mathbb {R}^3, \end{array} \right. \end{aligned}$$ where $$\varepsilon >0$$ is a small parameter, $$\frac{3}{4}0$$ sufficiently small, and we determine a concrete set related to the potentials V and K as the concentration position of these ground state solutions as $$\varepsilon \rightarrow 0$$ . Moreover, we considered some properties of these ground state solutions, such as convergence and decay estimate.

In this paper, we are concerned with the existence and concentration behavior of ground states for the following quasilinear problem with competing potentials − e 2 Δ u + V ( x ) u − e 2 1 2 Δ ( u 2 ) u = P ( x ) | u | p − 1 u + Q ( x ) | u | q − 1 u , where 3 q p 2 2 ∗ − 1 , 2 ∗ is the Sobolev critical exponent, V ( x ) and P ( x ) are positive and Q ( x ) may be sign-changing. We show the existence of the ground states via the Nehari manifold method for e > 0 , and these ground states “concentrate” at a global minimum point of the least energy function C ( s ) as e → 0 + .

We are concerned with the existence and concentration behavior of ground state solutions of the fractional Schrödinger–Poisson system with critical nonlinearity [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text], [Formula: see text] denotes the fractional Laplacian of order [Formula: see text] and satisfies [Formula: see text]. The potential [Formula: see text] is continuous and positive, and has a local minimum. We obtain a positive ground state solution for [Formula: see text] small, and we show that these ground state solutions concentrate around a local minimum of [Formula: see text] as [Formula: see text].

This paper is concerned with the existence and concentration properties of the ground state solutions to the following coupled Schrodinger systems $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)G_{v}(z)~\hbox { in }\ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)G_{u}(z)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)(G_{v}(z)+|z|^{2^*-2}v)~\hbox {in } \ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)(G_{u}(z)+|z|^{2^*-2}u)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where $$z=(u,v)\in {\mathbb {R}}^2$$ , $$G$$ is a power type nonlinearity, having superquadratic growth at both $$0$$ and infinity but subcritical, $$V$$ can be sign-changing and $$\inf W>0$$ . We prove the existence, exponential decay, $$H^2$$ -convergence and concentration phenomena of the ground state solutions for small $$\varepsilon >0$$ .

We consider the existence of infinitely many solutions of a class of quasilinear problems, for which the corresponding functionals are non-symmetric

In this paper, we consider the following system: where Ω ⊂ ℝN (N ≥ 3) is a smooth bounded domain and α, β, λ, μ > 1 are parameters with . By using the perturbation method, we prove the existence of positive solutions for the problem (P). Indeed, the method we are using in this paper can be applied to the more general quasilinear system (see (1.7)).

Eight-stranded β-barrel outer membrane proteins can confer bacterial virulence via resistance to host innate defenses. This resistance function of OmpW, which was recently identified as an eight-stranded β-barrel protein, was investigated in this study. Our results demonstrated that upregulation of OmpW correlated with increased bacterial survival during phagocytosis. Bacterial mutants harboring a deletion of ompW exhibited a significantly increased phagocytosis rate. Both observations suggest that the OmpW protein protects bacteria against host phagocytosis. In addition, expression of ompW is regulated by iron, which implies that the resistance provided by OmpW may be an important factor in iron-related infectious diseases. Furthermore, OmpW has been identified as a protective antigen that protects mice against bacterial infection and is therefore a promising target for vaccine development against infectious diseases.

In this paper we study a system of Schrodinger–Poisson equations { − Δ u + λ V ( x ) u + K ( x ) ϕ u = | u | p − 2 u , in R 3 , − Δ ϕ = K ( x ) u 2 , in R 3 , where λ > 0 is a parameter, 2 p 6 . Under suitable assumptions on V and K, the existence of nontrivial solution and concentration results are obtained via variational methods. In particular, the potential V is allowed to be sign-changing for the case p ∈ ( 4 , 6 ) .

This paper is concerned with the existence of the ground state solutions for the following superlinear elliptic system of Hamiltonian type, { − Δ u + V ( x ) u = g ( x , v ) in R N , − Δ v + V ( x ) v = f ( x , u ) in R N , u ( x ) → 0 and v ( x ) → 0 as | x | → ∞ , where V ∈ C ( R N , R ) is periodic in x 1 , x 2 , … , x N . We assume that 0 lies in a gap of the spectrum − Δ + V , and f and g are both superlinear at 0 and infinity but they have different increasing rates at infinity. By proving all Cerami sequences for the energy functional are bounded, existence of a ground state solution is obtained.

This paper is concerned with the following nonperiodic Hamiltonian elliptic system { − Δ u + V ( x ) u = H v ( x , u , v ) − Δ v + V ( x ) v = H u ( x , u , v ) x ∈ R N , u ( x ) , v ( x ) → 0 as | x | → ∞ , where z = ( u , v ) : R N → R × R , N ≥ 3 . Assuming the potential V is nonperiodic and sign-changing, H ( x , z ) is nonperiodic in x and asymptotically quadratic in z = ( u , v ) , existence and multiplicity of solutions are obtained via a variational approach.

This paper is concerned with the nonperiodic Dirac equations { − i ∑ k = 1 3 α k ∂ k u + a β u + M ( x ) u = F u ( x , u ) in R 3 , u ( x ) → 0 as | x | → ∞ , where M ( x ) is a vector potential. We introduce a new assumption on the potential M , and prove that the corresponding Dirac operator has only a point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of multiple solutions for asymptotically quadratic nonlinearity as well as the existence of infinitely many solutions for superquadratic nonlinearity.

In this paper, we study the following Hamiltonian elliptic systems $$\left\{\begin{array}{ll}-\Delta u+V(x)u= g(x,v),\quad {\rm in }\, \mathbb{R}^N,\\-\Delta v+V(x)v= f(x,u),\quad {\rm in } \, \mathbb{R}^N.\end{array}\right.$$ where $${V(x)\in C(\mathbb R^N), f(x,t), g(x,t)\in C(\mathbb{R}^N\times \mathbb{R})}$$ are superlinear in t at infinity. Without Ambrosetti–Rabinowtitz condition, the existences of ground state solutions are obtained via the combination of generalized linking theorem and monotonicity method.

Three isomers of Sm@C82 that are soluble in organic solvents were obtained from the carbon soot produced by vaporization of hollow carbon rods doped with Sm2O3/graphite powder in an electric arc. These isomers were numbered as Sm@C82(I), Sm@C82(II), and Sm@C82(III) in order of their elution times from HPLC chromatography on a Buckyprep column with toluene as the eluent. The identities of isomers, Sm@C82(I) as Sm@Cs(6)-C82, Sm@C82(II) as Sm@C3v(7)-C82, and Sm@C82(III) as Sm@C2(5)-C82, were determined by single-crystal X-ray diffraction on cocrystals formed with Ni(octaethylporphyrin). For endohedral fullerenes like La@C82, which have three electrons transferred to the cage to produce the M3+@(C82)3– electronic distribution, generally only two soluble isomers (e.g., La@C2v(9)-C82 (major) and La@Cs(6)-C82 (minor)) are observed. In contrast, with samarium, which generates the M2+@(C82)2– electronic distribution, five soluble isomers of Sm@C82 have been detected, three in this study, the other two in two relat...

This paper is concerned with the semiclassical nonlinear Schrodinger equation { − e 2 Δ u + V ( x ) u = f ( x , u ) , x ∈ R N u ( x ) → 0 as | x | → ∞ . where e > 0 is a small parameter. Under certain hypotheses on V and a general spectral assumption, the existence and multiplicity of semiclassical solutions are obtained for asymptotically linear nonlinearity via variational methods.

This paper is concerned with the following periodic Hamiltonian elliptic system $ -\Delta u+V(x)u=g(x,v)$ in $R^N,$ $ -\Delta v+V(x)v=f(x,u)$ in $R^N,$ $ u(x)\to 0$ and $v(x)\to 0$ as $|x|\to\infty,$ where the potential $V$ is periodic and has a positive bound from below, $f(x,t)$ and $g(x,t)$ are periodic in $x$ and superlinear but subcritical in $t$ at infinity. By using generalized Nehari manifold method, existence of a positive ground state solution as well as multiple solutions for odd $f$ and $g$ are obtained.

This paper is concerned with the following periodic Hamiltonian elliptic system $\-\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)$ in $\mathbb{R}^N,$ $\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)$ in $\mathbb{R}^N,$ $\varphi(x)\to 0$ and $\psi(x)\to0$ as $|x|\to\infty.$ Assuming the potential $V$ is periodic and $0$ lies in a gap of $\sigma(-\Delta+V)$, $G(x,\eta)$ is periodic in $x$ and superquadratic in $\eta=(\varphi,\psi)$, existence and multiplicity of solutions are obtained via variational approach.

This paper is concerned with the following Hamiltonian elliptic system { − Δ u + b ( x ) ⋅ ∇ u + V ( x ) u = H v ( x , u , v ) , − Δ v − b ( x ) ⋅ ∇ v + V ( x ) v = H u ( x , u , v ) for x ∈ R N . Existence and multiplicity of solutions are obtained for the systems with periodic or non-periodic potentials V via variational methods.

This paper is concerned with the following periodic Hamiltonian elliptic system $$\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right.$$ where the potential V is periodic and 0 lies in a gap of the spectrum of −Δ + V, f(x, t) and g(x, t) depend periodically on x and are superlinear but subcritical in t at infinity. By establishing a variational setting, existence of a ground state solution and multiple solution for odd f and g are obtained.

In this paper, we consider the existence and multiplicity of solutions for the following quasilinear Choquard equation: −Δu+V(x)u−uΔ(u2)=(|x|−μ*|u|p)|u|p−2u, x∈RN, where N ≥ 3, μ∈(0,N+22), p∈(2,4N−4μN−2). Under some assumptions on V, we obtain the existence of positive solutions, negative solutions, and high-energy solutions via perturbation method.