scholarly journals Existence and Multiplicity of Solutions for Semipositone Problems Involvingp-Laplacian

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Guowei Dai ◽  
Chunfeng Yang
Keyword(s):  
Positive Solutions ◽  
Smooth Domain ◽  
Multiplicity Of Solutions ◽  
Bounded Smooth Domain ◽  
Existence And Multiplicity ◽  
Bounded Smooth ◽  
Multiplicity Of Positive Solutions

We prove existence and multiplicity of positive solutions for semipositone problems involvingp-Laplacian in a bounded smooth domain ofℝNunder the cases of sublinear and superlinear nonlinearities term.

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.

scholarly journals High perturbations of a new Kirchhoff problem involving the p-Laplace operator

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhongyi Zhang ◽  
Yueqiang Song
Keyword(s):  
Laplace Operator ◽  
Smooth Domain ◽  
Multiplicity Of Solutions ◽  
Bounded Smooth Domain ◽  
Existence And Multiplicity ◽  
Bounded Smooth ◽  
Nonlocal Terms ◽  
Kirchhoff Problem

AbstractIn the present work we are concerned with the existence and multiplicity of solutions for the following new Kirchhoff problem involving the p-Laplace operator: $$ \textstyle\begin{cases} - (a-b\int _{\Omega } \vert \nabla u \vert ^{p}\,dx ) \Delta _{p}u = \lambda \vert u \vert ^{q-2}u + g(x, u), & x \in \Omega , \\ u = 0, & x \in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | p d x ) Δ p u = λ | u | q − 2 u + g ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $a, b > 0$ a , b > 0 , $\Delta _{p} u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplace operator, $1 < p < N$ 1 < p < N , $p < q < p^{\ast }:=(Np)/(N-p)$ p < q < p ∗ : = ( N p ) / ( N − p ) , $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N \geq 3$ N ≥ 3 ) is a bounded smooth domain. Under suitable conditions on g, we show the existence and multiplicity of solutions in the case of high perturbations (λ large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.

scholarly journals Sign-changing solutions for a Schrödinger–Kirchhoff–Poisson system with 4-sublinear growth nonlinearity

2021 ◽  
pp. 1-21
Author(s):  
Shubin Yu ◽  
Ziheng Zhang ◽  
Rong Yuan
Keyword(s):  
Type System ◽  
Invariant Sets ◽  
Smooth Domain ◽  
Positive Parameter ◽  
Poisson System ◽  
Bounded Smooth Domain ◽  
Existence And Multiplicity ◽  
Bounded Smooth ◽  
Sign Changing Solutions ◽  
Poisson Type

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system { − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u + λ ϕ u = Q ( x ) | u | p − 2 u in   Ω , − Δ ϕ = u 2 in   Ω , u = ϕ = 0 on   ∂ Ω , where Ω is a bounded smooth domain of R 3 , a > 0 , b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q ( x ) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of sign-changing solutions to this problem for the case that 2 < p < 4 as λ sufficient small. Furthermore, for λ = 1 and the above assumptions on Q ( x ) , we obtain the same conclusions with 2 < p < 12 5 .

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 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 the Existence of Solutions of a Nonlocal Elliptic Equation with ap-Kirchhoff-Type Term

2008 ◽  
Vol 2008 ◽  
pp. 1-25 ◽  
Author(s):  
Francisco Julio S. A. Corrêa ◽  
Rúbia G. Nascimento
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Kirchhoff Type ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth

Questions on the existence of positive solutions for the following class of elliptic problems are studied:−[M(‖u‖1,pp)]1,pΔpu=f(x,u), inΩ,u=0, on∂Ω, whereΩ⊂ℝNis a bounded smooth domain,f:Ω¯×ℝ+→ℝandM:ℝ+→ℝ,  ℝ+=[0,∞)are given functions.

scholarly journals Ground state solutions of p-Laplacian singular Kirchhoff problem involving a Riemann-Liouville fractional derivative

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2073-2088 ◽  
Author(s):  
Mouna Kratou
Keyword(s):  
Fractional Derivative ◽  
Variational Methods ◽  
Positive Solutions ◽  
Ground State ◽  
Multiplicity Of Solutions ◽  
Singular Nonlinearity ◽  
Existence And Multiplicity ◽  
Liouville Fractional Derivative ◽  
Multiplicity Of Positive Solutions ◽  
Kirchhoff Problem

The purpose of this paper is to study the existence and multiplicity of solutions to the following Kirchhoff equation with singular nonlinearity and Riemann-Liouville Fractional Derivative: (P?){a+b ?T0|0D?t(u(t))|pdt)p-1 tD?T (?p(0D?tu(t)) = ?g(t)/u?(t) + f(t, u(t)), t ? (0,T); u(0)=u(T)=0, where a ? 1, b, ? > 0, p > 1 are constants, 1/p < ? ? 1, 0 < ? < 1, g ? C([0,1]) and f ? C1([0,T] x R,R). Under appropriate assumptions on the function f, we employ variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter ?.

scholarly journals Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 107
Author(s):  
Daliang Zhao ◽  
Juan Mao
Keyword(s):  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Integral Boundary ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

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