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 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.

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>

scholarly journals Existence of Solutions for a Modified Nonlinear Schrödinger System

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yujuan Jiao ◽  
Yanli Wang
Keyword(s):  
Perturbation Method ◽  
Existence Of Solutions ◽  
Critical Sobolev Exponent ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrödinger System ◽  
Bounded Smooth ◽  
Sobolev Exponent ◽  
Schrödinger System

We are concerned with the following modified nonlinear Schrödinger system:-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu,  x∈Ω,  -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v,  x∈Ω,  u=0,  v=0,  x∈∂Ω, whereα>2,  β>2,  α+β<2·2*,  2*=2N/(N-2)is the critical Sobolev exponent, andΩ⊂ℝN  (N≥3)is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

Positive and nodal solutions for an elliptic equation with critical growth

2016 ◽  
Vol 18 (02) ◽  
pp. 1550021 ◽  
Author(s):  
Marcelo F. Furtado ◽  
Bruno N. Souza
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Critical Growth ◽  
Smooth Domain ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Bounded Smooth Domain ◽  
Continuous Potential ◽  
Bounded Smooth

We consider the problem [Formula: see text] where [Formula: see text] is a bounded smooth domain, [Formula: see text], [Formula: see text], [Formula: see text]. Under some suitable conditions on the continuous potential [Formula: see text] and on the parameter [Formula: see text], we obtain one nodal solution for [Formula: see text] and one positive solution for [Formula: see text].

scholarly journals Multiple Sign-Changing Solutions for Kirchhoff-Type Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Xingping Li ◽  
Xiumei He
Keyword(s):  
Mountain Pass ◽  
Smooth Domain ◽  
Three Solutions ◽  
Bounded Smooth Domain ◽  
Kirchhoff Type ◽  
Bounded Smooth ◽  
Sign Changing Solutions

We study the following Kirchhoff-type equations-a+b∫Ω∇u2dxΔu+Vxu=fx,u, inΩ,u=0, in∂Ω, whereΩis a bounded smooth domain ofRN  (N=1,2,3),a>0,b≥0,f∈C(Ω¯×R,R), andV∈C(Ω¯,R). Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, iffis odd with respect to its second variable, this problem has infinitely many sign-changing solutions.

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 Existence of positive solutions for a class of the p-Laplace equations

1994 ◽  
Vol 36 (2) ◽  
pp. 249-264 ◽  
Author(s):  
Yin Xi Huang
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solution ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Unique Positive Solution ◽  
Bounded Smooth Domain ◽  
Laplace Equations ◽  
Existence Of Positive Solutions ◽  
Bounded Smooth ◽  
Image Position

AbstractWe are concerned with the existence of solutions ofwhere Δp is the p-Laplacian, p ∈ (1, ∞), and Ω is a bounded smooth domain in ℝn.For h(x) ≡ 0 and f(x, u) satisfying proper asymptotic spectral conditions, existence of a unique positive solution is obtained by invoking the sub-supersolution technique and the spectral method. For h(x) ≢ 0, with assumptions on asymptotic behavior of f(x, u) as u → ±∞, an existence result is also proved.

scholarly journals Nonexistence of nontrivialweak solutions for anisotropic elliptic problems

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5415-5420
Author(s):  
Tingfu Feng
Keyword(s):  
Weak Solutions ◽  
Elliptic Problems ◽  
Smooth Domain ◽  
Bounded Smooth Domain ◽  
Size And Shape ◽  
Bounded Smooth

In this paper, we consider anisotropic elliptic problems on a bounded smooth domain, nonexistence of nontrivial weak solutions are obtained by two different simplified methods. We point out the results of nonexistence of nontrivial weak solutions not only depend on the size and shape of domain, but also the constants p1 and pN in anisotropic elliptic problems.

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.

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