scholarly journals Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jianqing Chen ◽  
Qian Zhang
Keyword(s):  
Soliton Solutions ◽  
Nodal Solution ◽  
Entire Space ◽  
Nodal Solutions ◽  
Moser Iteration ◽  
Nodal Domains ◽  
Alpha Beta ◽  
Beta 2 ◽  
Principle Of Symmetric Criticality ◽  
Schrödinger System

<p style='text-indent:20px;'>This paper is concerned with the following quasilinear Schrödinger system in the entire space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^{N}(N\geq3) $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{aligned} &amp;-\Delta u+A(x)u+\frac{k}{2}\triangle(u^{2})u = \frac{2\alpha }{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &amp;-\Delta v+Bv+\frac{k}{2}\triangle(v^{2})v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ &amp; u(x)\to 0,\ \ v(x)\to 0\ \ \hbox{as}\ |x|\to \infty,\end{aligned}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \alpha,\beta&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 2&lt;\alpha+\beta&lt;2^* = \frac{2N}{N-2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ k &gt;0 $\end{document}</tex-math></inline-formula> is a parameter. By using the principle of symmetric criticality and the moser iteration, for any given integer <inline-formula><tex-math id="M5">\begin{document}$ \xi\geq2 $\end{document}</tex-math></inline-formula>, we construct a non-radially symmetrical nodal solution with its <inline-formula><tex-math id="M6">\begin{document}$ 2\xi $\end{document}</tex-math></inline-formula> nodal domains. Our results can be looked on as a generalization to results by Alves, Wang and Shen (Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J. Differ. Equ. 259 (2015) 318-343).</p>

Nodal Solutions for a Quasilinear Elliptic Equation Involving the p-Laplacian and Critical Exponents

2018 ◽  
Vol 18 (1) ◽  
pp. 17-40
Author(s):  
Yinbin Deng ◽  
Shuangjie Peng ◽  
Jixiu Wang
Keyword(s):  
Elliptic Equations ◽  
Order Term ◽  
Quasilinear Elliptic Equations ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Content Type ◽  
Nodal Domains ◽  
Change Of Variables ◽  
Graphic Content ◽  
Quasilinear Elliptic

AbstractThis paper is concerned with the following type of quasilinear elliptic equations in{\mathbb{R}^{N}}involving thep-Laplacian and critical growth:-\Delta_{p}u+V(|x|)|u|^{p-2}u-\Delta_{p}(|u|^{2})u=\lambda|u|^{q-2}u+|u|^{2p^{% *}-2}u,which arises as a model in mathematical physics, where{2<p<N},{p^{*}=\frac{Np}{N-p}}. For any given integer{k\geq 0}, by using change of variables and minimization arguments, we obtain, under some additional assumptions onpandq, a radial sign-changing nodal solution with{k+1}nodal domains. Since the critical exponent appears and the lower order term (obtained by a transformation) may change sign, we shall use delicate arguments.

INFINITELY MANY NODAL SOLUTIONS FOR A WEAKLY COUPLED NONLINEAR SCHRÖDINGER SYSTEM

2008 ◽  
Vol 10 (05) ◽  
pp. 651-669 ◽  
Author(s):  
L. A. MAIA ◽  
E. MONTEFUSCO ◽  
B. PELLACCI
Keyword(s):  
Variational Methods ◽  
Nodal Solution ◽  
Radial Solutions ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nodal Solutions ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Weakly Coupled ◽  
Nonlinear Schrodinger Equations ◽  
Schrödinger System

Existence of radial solutions with a prescribed number of nodes is established, via variational methods, for a system of weakly coupled nonlinear Schrödinger equations. The main goal is to obtain a nodal solution with all vector components not identically zero and an estimate on their energies.

scholarly journals On sign-changing solutions for (p,q)-Laplace equations with two parameters

2016 ◽  
Vol 8 (1) ◽  
pp. 101-129 ◽  
Author(s):  
Vladimir Bobkov ◽  
Mieko Tanaka
Keyword(s):  
Dirichlet Problem ◽  
One Dimension ◽  
Linking Theorem ◽  
Nodal Solutions ◽  
Eigenvalues And Eigenfunctions ◽  
Nodal Domains ◽  
Laplace Equations ◽  
Alpha Beta ◽  
Two Parameters ◽  
Sign Changing Solutions

Abstract We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous {(p,q)} -Laplace equations {-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2% }u} where {p\neq q} . By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the {(\alpha,\beta)} -plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p- and q-Laplacians in one dimension.

Constant sign and nodal solutions for superlinear double phase problems

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Constant Sign ◽  
Nodal Solution ◽  
Negative Solution ◽  
Nontrivial Solutions ◽  
Nodal Solutions ◽  
Nodal Domains ◽  
Double Phase ◽  
Variational Tools ◽  
Nehari Method ◽  
Superlinear Reaction

Abstract We consider a double phase problems with unbalanced growth and a superlinear reaction, which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools and the Nehari method, we show that the Dirichlet problem has at least three nontrivial solutions, a positive solution, a negative solution and a nodal solution. The nodal solution has exactly two nodal domains.

scholarly journals Multiple nodal solutions for the Schr\”odinger-Poisson system with an asymptotically cubic term

2021 ◽  
Author(s):  
Hui Guo ◽  
Ronghua Tang ◽  
Tao Wang
Keyword(s):  
Positive Integer ◽  
Open Problem ◽  
Nonlinear Term ◽  
Affirmative Answer ◽  
Poisson System ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Nodal Domains ◽  
Deformation Lemma ◽  
Asymptotically Cubic

This paper deals with the following Schr\“odinger-Poisson system \begin{equation}\left\{\begin{aligned} &-\Delta u+u+ \lambda\phi u=f(u)\quad\mbox{in }\mathbb{R}^3,\\ &-\Delta \phi=u^{2}\quad\mbox{in }\mathbb{R}^3, \end{aligned}\right.\end{equation} where $\lambda>0$ and $f(u)$ is a nonlinear term asymptotically cubic at the infinity. Taking advantage of the Miranda theorem and deformation lemma, we combine some new analytic techniques to prove that for each positive integer $k,$ system \eqref{zhaiyaofc} admits a radial nodal solution $U_k^{\lambda}$, which has exactly $k+1$ nodal domains and the corresponding energy is strictly increasing in $k$. Moreover, for any sequence $\{\lambda_n\}\to 0_+$ as $n\to\infty,$ up to a subsequence, $U_k^{\lambda_n}$ converges to some $U_k^0\in H_r^1(\mathbb{R}^3)$, which is a radial nodal solution with exactly $k+1$ nodal domains of \eqref{zhaiyaofc} for $\lambda=0 $. These results give an affirmative answer to the open problem proposed in [Kim S, Seok J. Commun. Contemp. Math., 2012] for the Schr\”odinger-Poisson system with an asymptotically cubic term.

scholarly journals On alienation of two functional equations of quadratic type

2021 ◽  
Author(s):  
Roman Ger
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Real Numbers ◽  
Alpha Beta ◽  
Beta 2 ◽  
Special Case

Abstract  We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.

Projective flatness of a new class of ( α , β ) (\alpha,\beta) -metrics

2019 ◽  
Vol 26 (1) ◽  
pp. 133-139
Author(s):  
Laurian-Ioan Pişcoran ◽  
Vishnu Narayan Mishra
Keyword(s):  
Riemannian Metric ◽  
Real Scalar ◽  
New Class ◽  
Alpha Beta ◽  
Beta 2 ◽  
Projective Flatness ◽  
The Relationship

Abstract In this paper we investigate a new {(\alpha,\beta)} -metric {F=\beta+\frac{a\alpha^{2}+\beta^{2}}{\alpha}} , where {\alpha=\sqrt{{a_{ij}y^{i}y^{j}}}} is a Riemannian metric; {\beta=b_{i}y^{i}} is a 1-form and {a\in(\frac{1}{4},+\infty)} is a real scalar. Also, we investigate the relationship between the geodesic coefficients of the metric F and the corresponding geodesic coefficients of the metric α.

scholarly journals On Critical p-Laplacian Systems

2017 ◽  
Vol 17 (4) ◽  
pp. 641-659
Author(s):  
Zhenyu Guo ◽  
Kanishka Perera ◽  
Wenming Zou
Keyword(s):  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Existence And Multiplicity ◽  
Nonnegative Solutions ◽  
Least Energy Solution ◽  
Existence And Nonexistence ◽  
Alpha Beta ◽  
Beta 2 ◽  
Sobolev Exponent ◽  
Least Energy

AbstractWe consider the critical p-Laplacian system\left\{\begin{aligned} &\displaystyle{-}\Delta_{p}u-\frac{\lambda a}{p}\lvert u% \rvert^{a-2}u\lvert v\rvert^{b}=\mu_{1}\lvert u\rvert^{p^{\ast}-2}u+\frac{% \alpha\gamma}{p^{\ast}}\lvert u\rvert^{\alpha-2}u\lvert v\rvert^{\beta},&&% \displaystyle x\in\Omega,\\ &\displaystyle{-}\Delta_{p}v-\frac{\lambda b}{p}\lvert u\rvert^{a}\lvert v% \rvert^{b-2}v=\mu_{2}\lvert v\rvert^{p^{\ast}-2}v+\frac{\beta\gamma}{p^{\ast}}% \lvert u\rvert^{\alpha}\lvert v\rvert^{\beta-2}v,&&\displaystyle x\in\Omega,\\ &\displaystyle u,v\text{ in }D_{0}^{1,p}(\Omega),\end{aligned}\right.where {\Delta_{p}u:=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplacian operator defined onD^{1,p}(\mathbb{R}^{N}):=\bigl{\{}u\in L^{p^{\ast}}(\mathbb{R}^{N}):\lvert% \nabla u\rvert\in L^{p}(\mathbb{R}^{N})\bigr{\}},endowed with the norm {{\lVert u\rVert_{D^{1,p}}:=(\int_{\mathbb{R}^{N}}\lvert\nabla u\rvert^{p}\,dx% )^{\frac{1}{p}}}}, {N\geq 3}, {1<p<N}, {\lambda,\mu_{1},\mu_{2}\geq 0}, {\gamma\neq 0}, {a,b,\alpha,\beta>1} satisfy {a+b=p}, {\alpha+\beta=p^{\ast}:=\frac{Np}{N-p}}, the critical Sobolev exponent, Ω is {\mathbb{R}^{N}} or a bounded domain in {\mathbb{R}^{N}} and {D_{0}^{1,p}(\Omega)} is the closure of {C_{0}^{\infty}(\Omega)} in {D^{1,p}(\mathbb{R}^{N})}. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution of this system. We also consider the existence and multiplicity of the nontrivial nonnegative solutions.

A spectrin-like protein from mouse brain membranes: phosphorylation of the 235,000-dalton subunit

1984 ◽  
Vol 247 (1) ◽  
pp. C61-C73 ◽  
Author(s):  
S. R. Goodman ◽  
I. S. Zagon ◽  
C. F. Whitfield ◽  
L. A. Casoria ◽  
S. B. Shohet ◽  
...  
Keyword(s):  
Mouse Brain ◽  
Beta Subunit ◽  
Beta Subunits ◽  
Mouse Erythrocyte ◽  
Alpha Beta ◽  
Beta 2 ◽  
Independent Protein ◽  
Brain Spectrin

A mouse brain spectrin-like protein, which was an immunoreactive analogue of erythrocyte spectrin, has been isolated from demyelinated membranes. This spectrin analogue was a 10.5 S, 972,000 molecular weight (Mr) (alpha beta)2 tetramer containing subunits of 240,000 (alpha) and 235,000 (beta) Mr. We demonstrated that in vivo only the 235,000 Mr beta subunit of the mouse brain spectrin-like protein was phosphorylated, which was an analogous situation to mouse erythrocyte spectrin in which only the 220,000 Mr beta subunit was phosphorylated. Incubation of isolated membrane fractions with [gamma-32P]ATP +/- adenosine 3',5'-cyclic monophosphate (cAMP) indicated that mouse brain spectrin-like protein, mouse erythrocyte spectrin, and human erythrocyte spectrin's beta subunits were all phosphorylated in vitro by membrane-associated cAMP-independent protein kinases.

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