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1
Content available Some remarks on the optimization of eigenvalue problems involving the p-Laplacian
100%
Pielichowski W.
Opuscula Mathematica
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2008
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tom Vol. 28, no. 4
561-566
EN
Given a bounded domain Ω ⊂ Rn, numbers p > 1, ∝ ≥ 0 and A ∈ /0, /Ω/], consider the optimization problem: find a subset D ⊂ Ω, of measure A, for which the first eigenvalue of the operator u → - div(/∇u/p-2 ∇u) + ∝ΧD/u/p-2u with the Dirichlet boundary condition is as small as possible. We show that the optimal configuration D is connected with the corresponding positive eigenfunction u in such a way that there exists a number t ≥ 0 for which D = { u ≤ t}. We also give a new proof of symmetry of optimal solutions in the case when Ω is Steiner symmetric and p = 2.
2
Content available The first eigencurve for a Neumann boundary problem involving p-Laplacian with essentially bounded weights
100%
Sanhaji A. , Dakkak A. , Moussaoui M.
Opuscula Mathematica
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2023
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tom Vol. 43, no. 4
559--574
EN
This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation −Δp u = αm1|u|p−2u + βm2|u|p−2u in a bounded domain Ω ⊂ RN. We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability.
3
Content available remote Eigenvalue problems with indefinite weight
100%
Szulkin A. , Willem M.
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nr 2
191-201
EN
We consider the linear eigenvalue problem -Δu = λV(x)u, $u ∈ D^{1,2}_0(Ω)$, and its nonlinear generalization $-Δ_{p}u = λV(x)|u|^{p-2}u$, $u ∈ D^{1,p}_0(Ω)$. The set Ω need not be bounded, in particular, $Ω = ℝ^N$ is admitted. The weight function V may change sign and may have singular points. We show that there exists a sequence of eigenvalues $λ_n → ∞$.
4
Content available Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions
100%
Hai D. D. , Wang X.
Opuscula Mathematica
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2019
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tom Vol. 39, no. 5
675--689
EN
We prove the existence of positive solutions for the p-Laplacian problem [formula] where [formula] can be nonlinear, i=1,2 , f:(0,∞)→R is p-superlinear or p-sublinear at ∞ and is allowed be singular (±∞) at 0, and λ is a positive parameter.
5
Content available Properties of solutions to some weighted ρLlaplacian equation
100%
Garain P.
Opuscula Mathematica
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2020
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tom Vol. 40, no. 4
483--494
EN
In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by [formula] on smooth domain and for varying nonlinearity ∫.
6
Content available Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball
100%
Graef J. R. , Hebboul D. , Moussaoui T.
|
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tom Vol. 43, no. 1
47--66
EN
In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the p-Laplacian [formula] where λ > 0 is a parameter, Ωe = {x ∈ RN : |x| > r0}, r0 > 0, N > p > 1, Δp is the p-Laplacian operator, and f ∈ C([r0,+∞) × [0,+∞) ,R) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of λ.
7
Content available On the solvability of Dirichlet problem for the weighted p-Laplacian
100%
Szlachtowska E.
Opuscula Mathematica
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2012
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tom Vol. 32, no. 4
775-781
EN
The paper investigates the existence and uniqueness of weak solutions for a non-linear boundary value problem involving the weighted ρ-Laplacian. Our approach is based on variational principles and representation properties of the associated spaces.
8
Evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci-harmonic flow
89%
Abolarinwa A.
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2015
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tom Vol. 21, nr 2
147--160
EN
We study the evolution and monotonicity of the eigenvalues of p-Laplace operator on an m-dimen-sional compact Riemannian manifold M whose metric g(t) evolves by the Ricci-harmonic flow. The first nonzero eigenvalue is proved to be monotonically nondecreasing along the flow and differentiable almost everywhere. As a corollary, we recover the corresponding results for the usual Laplace-Beltrami operator when p = 2. We also examine the evolution and monotonicity under volume preserving flow and it turns out that the first eigenvalue is not monotone in general.
9
Comparison estimates on the first eigenvalue of a quasilinear elliptic system
89%
Abolarinwa A. , Azami S.
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2020
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tom Vol. 26, nr 2
273--285
EN
We study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and the inequality of Faber-Krahn for the first eigenvalue of a (p, q)-Laplacian are recovered. Lastly, we reprove a Cheeger-type estimate for the p-Laplacian, 1 < p < ∞, from where a lower bound estimate in terms of Cheeger’s constant for the first eigenvalue of a (p, q)-Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger’s constant as p, q → 1, 1.
10
Content available remote Energy decay result for a nonlinear wave p-Laplace equation with a delay term
89%
Zennir K. , Laouar L. K.
Mathematica Applicanda
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2017
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tom Vol. 45, No. 1
65--80
EN
We consider the nonlinear (in space and time) wave equation with delay term in the internal feedback. Under conditions on the delay term and the term without delay, we study the asymptotic behavior of solutions using the multiplier method and general weighted integral inequalities.
PL
Rozważamy nieliniowe równanie falowe (w czasie i przestrzeni) z członem wewnętrznego sprzężenia zwrotnego. Przy pewnych warunkach na poszczególne człony równania badane jest asymptotyczne zachowanie rozwiązań.
11
Content available remote Existence of solutions for a nonlinear problem at resonance
89%
Haddaoui M. , Lebrimchi H. , Ouhamou B. , Tsouli N.
Demonstratio Mathematica
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2022
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tom Vol. 55, nr 1
482--489
EN
In this work, we are interested at the existence of nontrivial solutions for a nonlinear elliptic problem with resonance part and nonlinear boundary conditions. Our approach is variational and is based on the well-known Landesman-Laser-type conditions.
12
Positive solutions for discrete boundary value problems with p-Laplacian
89%
Ji D. , Ge W.
Journal of Applied Analysis
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2010
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tom Vol. 16, nr 1
107-119
EN
In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following two-point discrete boundary value problem with p-Laplacian: Δ(∅p(Δu(k - 1))) + e(k)f(u(k)) = 0 , k∈ N(1,T) , u(0) - B0 (Δu(0)) = 0 , u(T + 1) + B1 (Δu(T)) = 0. The main tool is the monotone iterative technique.
13
Content available remote A generalized periodic boundary value problem for the one-dimensional p-Laplacian
88%
Jiang D. , Wang J.
|
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tom 65
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nr 3
265-270
EN
The generalized periodic boundary value problem -[g(u')]' = f(t,u,u'), a < t < b, with u(a) = ξu(b) + c and u'(b) = ηu'(a) is studied by using the generalized method of upper and lower solutions, where ξ,η ≥ 0, a, b, c are given real numbers, $g(s) = |s|^{p-2} s$, p > 1, and f is a Carathéodory function satisfying a Nagumo condition. The problem has a solution if and only if there exists a lower solution α and an upper solution β with α(t) ≤ β(t) for a ≤ t ≤ b.
14
Content available On the existence of three solutions for quasilinear elliptic problem
75%
Goncerz P.
Opuscula Mathematica
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2012
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tom Vol. 32, no. 3
473-486
EN
We consider a quasilinear elliptic problem of the type - Δpu = λ (ƒ (u)+ μg(u)) in Ω, u/∂Ω = 0, where Ω ⊂ RN is an open and bounded set, ƒ, g are continuous real functions on R and , λ, μ ∈ R. We prove the existence of at least three solutions for this problem using the so called three critical points theorem due to Ricceri.
15
Content available remote Existence and multiplicity results for nonlinear eigenvalue problems with discontinuities
75%
Papageorgiou N. S. , Papalini F.
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2000
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tom 75
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nr 2
125-141
EN
We study eigenvalue problems with discontinuous terms. In particular we consider two problems: a nonlinear problem and a semilinear problem for elliptic equations. In order to study the existence of solutions we replace these two problems with their multivalued approximations and, for the first problem, we estabilish an existence result while for the second problem we prove the existence of multiple nontrivial solutions. The approach used is variational.
16
Content available remote The existence of solution for boundary value problems for differential equations with deviating arguments and p-Laplacian
75%
Liu B. , Yu J.
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2000
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tom 75
|
nr 3
271-280
EN
We consider a boundary value problem for a differential equation with deviating arguments and p-Laplacian: $-(ϕ_{p}(x'))' + d/dt grad F(x) + g(t,x(t),x(δ(t))$, x'(t), x'(τ(t))) = 0, t ∈ [0,1]; $x(t)=\underline{φ}(t),$ t ≤ 0; $x(t) = \overline{φ}(t)$, t ≥ 1. An existence result is obtained with the help of the Leray-Schauder degree theory, with no restriction on the damping forces d/dt grad F(x).
17
Content available Periodic solutions for second-order Hamiltonian systems with a p-Laplacian
75%
Zhang X. , Tang X.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2010
|
tom 64
|
nr 1
EN
In this paper, by using the least action principle, Sobolev’s inequality and Wirtinger’s inequality, some existence theorems are obtained for periodic solutions of second-order Hamiltonian systems with a p-Laplacian under subconvex condition, sublinear growth condition and linear growth condition. Our results generalize and improve those in the literature.
18
Content available On the existence of five nontrivial solutions for resonant problems with p-Laplacian
75%
Gasiński L. , Papageorgiou N.
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tom 30
|
nr 2
169-189
EN
In this paper we study a nonlinear Dirichlet elliptic differential equation driven by the p-Laplacian and with a nonsmooth potential. The hypotheses on the nonsmooth potential allow resonance with respect to the principal eigenvalue λ₁ > 0 of $(-Δₚ,W₀^{1,p}(Z))$. We prove the existence of five nontrivial smooth solutions, two positive, two negative and the fifth nodal.
19
Content available remote Multiple solutions for nonlinear discontinuous elliptic problems near resonance
51%
Kourogenis N. C. , Papageorgiou N. S.
Colloquium Mathematicum
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1999
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tom 81
|
nr 1
89-99
EN
We consider a quasilinear elliptic eigenvalue problem with a discontinuous right hand side. To be able to have an existence theory, we pass to a multivalued problem (elliptic inclusion). Using a variational approach based on the critical point theory for locally Lipschitz functions, we show that we have at least three nontrivial solutions when $λ → λ_1$ from the left, $λ_1$ being the principal eigenvalue of the p-Laplacian with the Dirichlet boundary conditions.
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