scholarly journals Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Caixia Chen ◽  
Aixia Qian
Keyword(s):  
Poisson Equation ◽  
Positive Solutions ◽  
Ground State Solution ◽  
Multiple Positive Solutions ◽  
Bounded Smooth Domain ◽  
Nonlinear Growth ◽  
Concentration Compactness Principle ◽  
Spatial Dimensions ◽  
Bounded Smooth ◽  
Sobolev Critical Exponent

<p style='text-indent:20px;'>In this paper, we consider the following Schrödinger-Poisson equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{aligned} &amp;-\triangle u + u + \phi u = u^{5}+\lambda g(u), &amp;\hbox{in}\ \ \Omega, \\\ &amp; -\triangle \phi = u^{2}, &amp; \hbox{in}\ \ \Omega, \\\ &amp; u, \phi = 0, &amp; \hbox{on}\ \ \partial\Omega.\end{aligned}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded smooth domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{3} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> and the nonlinear growth of <inline-formula><tex-math id="M4">\begin{document}$ u^{5} $\end{document}</tex-math></inline-formula> reaches the Sobolev critical exponent in three spatial dimensions. With the aid of variational methods and the concentration compactness principle, we prove the problem admits at least two positive solutions and one positive ground state solution.</p>

Multiple Positive Solutions to a Kirchhoff Type Problem Involving a Critical Nonlinearity in ℝ3

2017 ◽  
Vol 17 (4) ◽  
pp. 661-676 ◽  
Author(s):  
Xiao-Jing Zhong ◽  
Chun-Lei Tang
Keyword(s):  
Positive Solutions ◽  
Principal Eigenvalue ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Critical Nonlinearity ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Kirchhoff Type Problem ◽  
The Nehari Manifold ◽  
Kirchhoff Type Problems

AbstractIn this paper, we investigate a class of Kirchhoff type problems in {\mathbb{R}^{3}} involving a critical nonlinearity, namely,-\biggl{(}1+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,dx\biggr{)}\triangle u=% \lambda f(x)u+|u|^{4}u,\quad u\in D^{1,2}(\mathbb{R}^{3}),where {b>0}, {\lambda>\lambda_{1}} and {\lambda_{1}} is the principal eigenvalue of {-\triangle u=\lambda f(x)u}, {u\in D^{1,2}(\mathbb{R}^{3})}. We prove that there exists {\delta>0} such that the above problem has at least two positive solutions for {\lambda_{1}<\lambda<\lambda_{1}+\delta}. Furthermore, we obtain the existence of ground state solutions. Our tools are the Nehari manifold and the concentration compactness principle. This paper can be regarded as an extension of Naimen’s work [21].

scholarly journals On a nonlinear system arising in a theory of thermal explosion

2018 ◽  
Vol 38 (2) ◽  
pp. 167-172
Author(s):  
S. H. Rasouli
Keyword(s):  
Mathematical Model ◽  
Nonlinear System ◽  
Thermal Explosion ◽  
Positive Solutions ◽  
Laplacian Operator ◽  
Multiplicity Results ◽  
Bounded Smooth Domain ◽  
Existence And Multiplicity ◽  
Bounded Smooth ◽  
Multiplicity Of Positive Solutions

The purpose of this paper is to study the existence and multiplicity of positive solutions for a mathematical model of thermal explosion which is described by the system$$\left\{\begin{array}{ll}-\Delta u = \lambda f(v), & x\in \Omega,\\-\Delta v = \lambda g(u), & x\in \Omega,\\\mathbf{n}.\nabla u+ a(u) u=0 , & x\in\partial \Omega,\\\mathbf{n}.\nabla v+ b(v) v=0 , & x\in\partial \Omega,\\\end{array}\right.$$where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N},$ $\Delta$ is the Laplacian operator, $\lambda>0$ is a parameter, $f,g$ belong to a class of non-negative functions that have a combined sublinear effect at $\infty,$ and $a,b: [0,\infty) \rightarrow (0,\infty)$ are nondecreasing $C^{1}$ functions. We establish our existence and multiplicity results by the method of sub-- and supersolutions.

Existence of positive solutions for nonlocal p ⁢ ( x ) p(x) -Kirchhoff elliptic systems

2019 ◽  
Vol 10 (1) ◽  
pp. 17-25 ◽  
Author(s):  
Salah Boulaaras ◽  
Rafik Guefaifia ◽  
Khaled Zennir
Keyword(s):  
Continuous Function ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth

Abstract In this article, we discuss the existence of positive solutions by using sub-super solutions concepts of the following {p(x)} -Kirchhoff system: \left\{\begin{aligned} &\displaystyle{-}M(I_{0}(u))\triangle_{p(x)}u=\lambda^{% p(x)}[\lambda_{1}f(v)+\mu_{1}h(u)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle{-}M(I_{0}(v))\triangle_{p(x)}v=\lambda^{p(x)}[\lambda_{2}g(u)+% \mu_{2}\tau(v)]&&\displaystyle\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\text{on }\partial\Omega,\end{aligned}\right. where {\Omega\subset\mathbb{R}^{N}} is a bounded smooth domain with {C^{2}} boundary {\partial\Omega} , {\triangle_{p(x)}u=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)} , {p(x)\in C^{1}(\overline{\Omega})} , with {1<p(x)} , is a function satisfying {1<p^{-}=\inf_{\Omega}p(x)\leq p^{+}=\sup_{\Omega}p(x)<\infty} , λ, {\lambda_{1}} , {\lambda_{2}} , {\mu_{1}} and {\mu_{2}} are positive parameters, {I_{0}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx} , and {M(t)} is a continuous function.

On a Class of Infinite Semipositone Nonlinear Systems with Multiple Parameters

2017 ◽  
Vol 84 (1-2) ◽  
pp. 90
Author(s):  
S. H. Rasouli
Keyword(s):  
Nonlinear Systems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Multiple Parameters ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth ◽  
Decreasing Functions

<p>We analyze the existence of positive solutions of infinite semipositone nonlinear systems with multiple parameters of the form</p><span>{</span>Δu = α<sub>1</sub> (f (v)) - 1/<sub>u</sub><sup>n</sup>) + β<sub>1</sub>(h (u) - 1/<sub>u</sub><sup>n</sup>),     x € Ω),<br /> -Δv = α<sub>2</sub> (g (u)) - 1/<sub>v</sub><sup>θ</sup>) + β<sub>2</sub>(k (v) - 1/<sub>u</sub><sup>θ</sup>),    x € Ω), <br /> u = v = 0,                                                x € δΩ),<p>where Ω is a bounded smooth domain of R<sup>N</sup>, η, θ ε (0, 1), and α<sub>1</sub>, α<sub>2</sub>, β<sub>1</sub> and β<sub>2</sub> are nonnegative parameters. Here f, g, h, k ε C ([0, ∞ ]), are non-decreasing functions and f(0), g(0), h(0), k(0) &gt; 0. We use the method of sub-super solutions to prove the existence of positive solution for α<sub>1</sub> + β<sub>1</sub> and α<sub>2</sub> + β<sub>2</sub> large.</p>

Structure of non-trivial non-negative solutions to singularly perturbed semilinear Dirichlet problems

2003 ◽  
Vol 133 (2) ◽  
pp. 363-392 ◽  
Author(s):  
Zongming Guo
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Singularly Perturbed ◽  
Smooth Domain ◽  
Dirichlet Problems ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

The structure of non-trivial non-negative solutions to singularly perturbed semilinear Dirichlet problems of the form −ε2Δu = f(u) in Ω, u = 0 on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied as ε → 0+, for a class of nonlinearities f(u) satisfying f(0) = f(z1) = f(z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1), f > 0 in (z1, z2) and . It is shown that there are many non-trivial non-negative solutions and they are spike-layer solutions. Moreover, the measure of each spike layer is estimated as ε → 0+. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0, ∞). Uniqueness of a large positive solution and many positive intermediate spike-layer solutions are obtained for ε sufficiently small.

scholarly journals On an elliptic equation of p-Kirchhoff type via variational methods

2006 ◽  
Vol 74 (2) ◽  
pp. 263-277 ◽  
Author(s):  
Francisco Júlio ◽  
S. A. Corrêa ◽  
Giovany M. Figueiredo
Keyword(s):  
Elliptic Equation ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Nonlocal Boundary ◽  
Continuous Functions ◽  
Bounded Smooth Domain ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth

This paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the p-Kirchhoff type and where Ω is a bounded smooth domain of ℝN, 1 < p < N, s ≥ p* = (pN)/(N – p) and M and f are continuous functions.

Asymptotic behaviour and symmetry of positive solutions to nonlinear elliptic equations in a half-space

2016 ◽  
Vol 146 (6) ◽  
pp. 1243-1263 ◽  
Author(s):  
Lei Wei
Keyword(s):  
Half Space ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Unique Positive Solution ◽  
Bounded Smooth Domain ◽  
Nonlinear Elliptic ◽  
Bounded Smooth ◽  
Image Position ◽  
General Nonlinearity

We consider the following equation:where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up(p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up. Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x1 only, and W satisfies

scholarly journals Multiplicity of Positive Solutions for ap-q-Laplacian Type Equation with Critical Nonlinearities

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Positive Solutions ◽  
Type Equation ◽  
Critical Sobolev Exponent ◽  
Critical Nonlinearity ◽  
Bounded Smooth Domain ◽  
Laplacian Equation ◽  
Critical Nonlinearities ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Multiplicity Of Positive Solutions

We study the effect of the coefficientf(x)of the critical nonlinearity on the number of positive solutions for ap-q-Laplacian equation. Under suitable assumptions forf(x)andg(x), we should prove that for sufficiently smallλ>0, there exist at leastkpositive solutions of the followingp-q-Laplacian equation,-Δpu-Δqu=fxu|p*-2u+λgxu|r-2u  in  Ω,u=0  on  ∂Ω,whereΩ⊂RNis a bounded smooth domain,N>p,1<q<N(p-1)/(N-1)<p≤max⁡{p,p^*-q/(p-1)}<r<p^*,p^*=Np/(N-p)is the critical Sobolev exponent, andΔsu=div(|∇u|s-2∇uis thes-Laplacian ofu.

scholarly journals On isolated singularities of Kirchhoff equations

2020 ◽  
Vol 10 (1) ◽  
pp. 102-120
Author(s):  
Huyuan Chen ◽  
Mouhamed Moustapha Fall ◽  
Binling Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Unique Solution ◽  
Positive Solutions ◽  
Point Theorem ◽  
Singular Solutions ◽  
Kirchhoff Equations ◽  
Subcritical Case ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

Abstract In this note, we study isolated singular positive solutions of Kirchhoff equation $$\begin{array}{} \displaystyle M_\theta(u)(-{\it\Delta}) u =u^p \quad{\rm in}\quad {\it\Omega}\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial {\it\Omega}, \end{array}$$ where p > 1, θ ∈ ℝ, Mθ(u) = θ + ∫Ω |∇ u| dx, Ω is a bounded smooth domain containing the origin in ℝN with N ≥ 2. In the subcritical case: 1 < p < $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ if N ≥ 3, 1 < p < + ∞ if N = 2, we employee the Schauder fixed point theorem to derive a sequence of positive isolated singular solutions for the above equation such that Mθ(u) > 0. To estimate Mθ(u), we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that Mθ(u) < 0, by analyzing relationship between the parameter λ and the unique solution uλ of $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda u^p=k\delta_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0). \end{array}$$ In the supercritical case: $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ ≤ p < $\begin{array}{} \displaystyle \frac{N+2}{N-2} \end{array}$ with N ≥ 3, we obtain two isolated singular solutions ui with i = 1, 2 such that Mθ(ui) > 0 under other assumptions.

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