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1
Content available The first eigencurve for a Neumann boundary problem involving p-Laplacian with essentially bounded weights
Sanhaji Ahmed, Dakkak Ahmed, Moussaoui Mimoun
Opuscula Mathematica
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2023
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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.
2
Content available Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball
Graef John R., Hebboul Doudja, Moussaoui Toufik
Opuscula Mathematica
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2023
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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.
3
Content available remote Existence of solutions for a nonlinear problem at resonance
Haddaoui Mustapha, Lebrimchi Hafid, Ouhamou Brahim, Tsouli Najib
Demonstratio Mathematica
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2022
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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.
4
Content available Properties of solutions to some weighted ρLlaplacian equation
Garain Prashanta
Opuscula Mathematica
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2020
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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 ∫.
5
Comparison estimates on the first eigenvalue of a quasilinear elliptic system
Abolarinwa Abimbola, Azami Shahroud
Journal of Applied Analysis
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2020
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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.
6
Content available Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions
Hai D. D., Wang X.
Opuscula Mathematica
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2019
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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.
7
Content available remote Energy decay result for a nonlinear wave p-Laplace equation with a delay term
Zennir K., Laouar L. K.
Mathematica Applicanda
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2017
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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.
8
Evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci-harmonic flow
Abolarinwa A.
Journal of Applied Analysis
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2015
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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
Content available remote Three Solutions Theorem for a Quasilinear Dirichlet Boundary Value Problem
Goncerz P.
Schedae Informaticae
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2012
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Vol. 21
159--168
EN
We consider a Dirichlet boundary value problem driven by the p-Laplacian with the right hand side being a Carathéodory function. The existence of solutions is obtained by the use of a special form of the three critical points theorem.
10
Content available On the solvability of Dirichlet problem for the weighted p-Laplacian
Szlachtowska E.
Opuscula Mathematica
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2012
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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.
11
Content available On the existence of three solutions for quasilinear elliptic problem
Goncerz P.
Opuscula Mathematica
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2012
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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.
12
Positive solutions for discrete boundary value problems with p-Laplacian
Ji D., Ge W.
Journal of Applied Analysis
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2010
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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 Some remarks on the optimization of eigenvalue problems involving the p-Laplacian
Pielichowski W.
Opuscula Mathematica
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2008
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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.
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