Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the p-Laplacian

2002 ◽  
Vol 132 (3) ◽  
pp. 511-519 ◽  
Author(s):  
Giovanni Anello ◽  
Giuseppe Cordaro
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Weak Solutions ◽  
Smooth Boundary ◽  
Open Set ◽  
Carathéodory Function ◽  
Image Position

In this paper we present a result of existence of infinitely many arbitrarily small positive solutions to the following Dirichlet problem involving the p-Laplacian, where Ω ∈ RN is a bounded open set with sufficiently smooth boundary ∂Ω, p > 1, λ > 0, and f: Ω × R → R is a Carathéodory function satisfying the following condition: there exists t̄ > 0 such that Precisely, our result ensures the existence of a sequence of a.e. positive weak solutions to the above problem, converging to zero in L∞(Ω).

Three positive solutions for semilinear elliptic problems involving concave and convex nonlinearities

2012 ◽  
Vol 142 (1) ◽  
pp. 115-135 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-li Lin
Keyword(s):  
Dirichlet Problem ◽  
Continuous Function ◽  
Positive Solutions ◽  
Smooth Boundary ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Concave And Convex Nonlinearities ◽  
Multiplicity Of Positive Solutions ◽  
Image Position

We study the existence and multiplicity of positive solutions for the Dirichlet problemwhere λ > 0, 1 < q < 2, p = 2* = 2N/(N − 2), 0 ε Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂Ω and f is a non-negative continuous function on $\bar{\varOmega}$. Assuming that f satisfies some hypothesis, we prove that the equation admits at least three positive solutions for sufficiently small λ.

Non-Local Elliptic Boundary-Value Problems

1968 ◽  
Vol 20 ◽  
pp. 1365-1382 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Differential Operators ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Operators ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Non Local ◽  
Image Position

Let G be a bounded open set of Rn with a smooth boundary ∂G. We consider the following elliptic boundary-value problem:where A and Bj are, respectively singular integro-differential operators on G and on ∂G, of orders 2m and rj with rj < 2m; Ck are boundary differential operators, and Ljk are linear operators, bounded in a sense to be specified.

A Quasi-Linear Elliptic Boundary Value Problem

1966 ◽  
Vol 18 ◽  
pp. 1105-1112 ◽  
Author(s):  
R. A. Adams
Keyword(s):  
Dirichlet Problem ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Classical Solutions ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Open Set ◽  
Image Position ◽  
Linear Elliptic Boundary ◽  
Typical Point

Let Ω be a bounded open set in Euclidean n-space, En. Let α = (α1, … , an) be an n-tuple of non-negative integers;and denote by Qm the set ﹛α| 0 ⩽ |α| ⩽ m}. Denote by x = (x1, … , xn) a typical point in En and putIn this paper we establish, under certain circumstances, the existence of weak and classical solutions of the quasi-linear Dirichlet problem1

Positive solutions of certain elliptic systems with density-dependent diffusions

1995 ◽  
Vol 125 (5) ◽  
pp. 1031-1050 ◽  
Author(s):  
Inkyung Ahn ◽  
Lige Li
Keyword(s):  
Positive Solutions ◽  
Growth Rates ◽  
Smooth Boundary ◽  
Elliptic Systems ◽  
Biological Interactions ◽  
Bounded Region ◽  
Density Dependent ◽  
Existence Of Positive Solutions ◽  
Image Position ◽  
Increasing Functions

Results are obtained on the existence of positive solutions to the following elliptic system:in a bounded region Ω in Rn with a smooth boundary, where the diffusion terms φ ψ are non-negative functions and the system could be degenerate, β γ are strictly increasing functions, k,σ ≧ 0 are constants. We assume also that the growth rates f, g satisfy certain monotonicities. Applications to biological interactions with density-dependent diffusions are given.

scholarly journals Bifurcation for some quasilinear operators

2001 ◽  
Vol 131 (4) ◽  
pp. 733-765 ◽  
Author(s):  
David Arcoya ◽  
José Carmona ◽  
Benedetta Pellacci
Keyword(s):  
Positive Solutions ◽  
Bifurcation Theory ◽  
Smooth Boundary ◽  
Elliptic Boundary ◽  
Complete Analysis ◽  
Multiplicity Results ◽  
Elliptic Boundary Value ◽  
The Matrix ◽  
Image Position ◽  
Quasilinear Elliptic

This paper deals with existence, uniqueness and multiplicity results of positive solutions for the quasilinear elliptic boundary-value problem , where Ω is a bounded open domain in RN with smooth boundary. Under suitable assumptions on the matrix A(x, s), and depending on the behaviour of the function f near u = 0 and near u = +∞, we can use bifurcation theory in order to give a quite complete analysis on the set of positive solutions. We will generalize in different directions some of the results in the papers by Ambrosetti et al., Ambrosetti and Hess, and Artola and Boccardo.

Uniqueness of positive solutions of a quasilinear Dirichlet problem with exponential nonlinearity

1998 ◽  
Vol 128 (5) ◽  
pp. 895-906 ◽  
Author(s):  
Adimurthi
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Exponential Nonlinearity ◽  
Unit Radius ◽  
Image Position

Uniqueness of positive solutions of the following equation:has been obtained, where B ⊂ ℝn is the ball of unit radius and λ > 0. Moreover, if n = 2, then the solutions are nondegenerate.

scholarly journals Nehari manifold and multiplicity result for elliptic equation involving p-laplacian problems

2018 ◽  
Vol 36 (4) ◽  
pp. 197-208
Author(s):  
Khaled Ben Ali ◽  
Abdeljabbar Ghanmi
Keyword(s):  
Elliptic Equation ◽  
Positive Solutions ◽  
Smooth Function ◽  
Smooth Boundary ◽  
Nehari Manifold ◽  
Multiplicity Result ◽  
Existence And Multiplicity ◽  
Open Set ◽  
Multiplicity Of Positive Solutions

This article shows the existence and multiplicity of positive solutions of the $p$-Laplacien problem $$\displaystyle -\Delta_{p} u=\frac{1}{p^{\ast}}\frac{\partial F(x,u)}{\partial u} + \lambda a(x)|u|^{q-2}u \quad \mbox{for } x\in\Omega;\quad \quad u=0,\quad \mbox{for } x\in\partial\Omega$$ where $\Omega$ is a bounded open set in $\mathbb{R}^n$ with smooth boundary, $1<q<p<n$, $p^{\ast}=\frac{np}{n-p}$, $\lambda \in \mathbb{R}\backslash \{0\}$ and $a$ is a smooth function which may change sign in $\overline{\Omega}$. The method is based on Nehari results on three sub-manifolds of the space $W_{0}^{1,p}$.

Multiple positive solutions for a nonlinear Dirichlet problem with non-convex vector-valued response

2005 ◽  
Vol 135 (1) ◽  
pp. 105-117 ◽  
Author(s):  
Andrzej Nowakowski ◽  
Andrzej Rogowski
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Multiple Positive Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Image Position ◽  
Vector Valued

In this paper we establish the existence of many positive solutions to with Vx a vector-valued nonlinearity.

A perturbation result of semilinear elliptic equations in exterior strip domains

1997 ◽  
Vol 127 (5) ◽  
pp. 983-1004 ◽  
Author(s):  
Tsing-san Hsu ◽  
Hwai-chiuan Wang
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Energy Solution ◽  
Perturbation Result ◽  
Semilinear Elliptic ◽  
Image Position

SynopsisIn this paper we show that if the decay of nonzero ƒ is fast enough, then the perturbation Dirichlet problem −Δu + u = up + ƒ(z) in Ω has at least two positive solutions, wherea bounded C1,1 domain S = × ω Rn, D is a bounded C1,1 domain in Rm+n such that D ⊂⊂ S and Ω = S\D. In case ƒ ≡ 0, we assert that there is a positive higher-energy solution providing that D is small.

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
Robert Černý
Keyword(s):  
Dirichlet Problem ◽  
Continuous Function ◽  
Small Parameter ◽  
Orlicz Space ◽  
Weak Solutions ◽  
Critical Growth ◽  
Young Function ◽  
Bounded Set ◽  
Open Set ◽  
Continuous Potential

AbstractLet n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem $$u \in W_0^1 L^\Phi \left( \Omega \right) and - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}} {{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u} {{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega ,$$ where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω = ℝn.

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