scholarly journals MULTIPLE SOLUTIONS FOR A DIRICHLET PROBLEM WITH p-LAPLACIAN AND SET-VALUED NONLINEARITY

2008 ◽  
Vol 77 (2) ◽  
pp. 285-303 ◽  
Author(s):  
D. AVERNA ◽  
S. A. MARANO ◽  
D. MOTREANU
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Multiple Solutions ◽  
Negative Solution ◽  
Dirichlet Eigenvalue ◽  
Variational Techniques ◽  
Nonsmooth Functions ◽  
Dirichlet Eigenvalue Problem

AbstractThe existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via sub- and supersolution methods as well as variational techniques for nonsmooth functions.

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

scholarly journals Sign-changing and multiple solutions for thep-Laplacian

2002 ◽  
Vol 7 (12) ◽  
pp. 613-625 ◽  
Author(s):  
Siegfried Carl ◽  
Kanishka Perera
Keyword(s):  
Positive Solution ◽  
Multiple Solutions ◽  
Negative Solution ◽  
Jumping Nonlinearities

We obtain a positive solution, a negative solution, and a sign-changing solution for a class ofp-Laplacian problems with jumping nonlinearities using variational and super-subsolution methods.

scholarly journals Eigenvalues for a combination between local and nonlocal p-Laplacians

2019 ◽  
Vol 22 (5) ◽  
pp. 1414-1436 ◽  
Author(s):  
Leandro M. Del Pezzo ◽  
Raúl Ferreira ◽  
Julio D. Rossi
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Limit Problem ◽  
First Eigenvalue ◽  
Order Zero ◽  
Dirichlet Eigenvalue ◽  
The First Eigenvalue ◽  
Homogeneous Operator ◽  
Dirichlet Eigenvalue Problem

Abstract In this paper we study the Dirichlet eigenvalue problem $$\begin{array}{} \displaystyle -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \text{ in } \Omega,\quad u=0 \, \text{ in } \, \Omega^c=\mathbb{R}^N\setminus\Omega. \end{array}$$ Here Ω is a bounded domain in ℝN, Δpu is the standard local p-Laplacian and ΔJ,pu is a nonlocal p-homogeneous operator of order zero. We show that the first eigenvalue (that is isolated and simple) satisfies $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$(λ1)1/p = Λ where Λ can be characterized in terms of the geometry of Ω. We also find that the eigenfunctions converge, u∞ = $\begin{array}{} \displaystyle \lim_{p\to\infty} \end{array}$ up, and find the limit problem that is satisfied in the limit.

scholarly journals The Ratio of Eigenvalues of the Dirichlet Eigenvalue Problem for Equations with One-Dimensionalp-Laplacian

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Gabriella Bognár ◽  
Ondřej Došlý
Keyword(s):  
Eigenvalue Problem ◽  
Dirichlet Eigenvalue ◽  
One Dimensional ◽  
Single Well ◽  
Dirichlet Eigenvalue Problem

We establish an estimate for the ratio of eigenvalues of the Dirichlet eigenvalue problem for the equation with one-dimensionalp-Laplacian involving a nonnegative unimodal (single-well) potential.

Multiple solutions for semilinear elliptic equations in unbounded cylinder domains

2004 ◽  
Vol 134 (4) ◽  
pp. 719-731 ◽  
Author(s):  
Tsing-San Hsu
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
Unbounded Cylinder ◽  
Image Position

In this paper, we show that if b(x) ≥ b∞ > 0 in Ω̄ and there exist positive constants C, δ, R0 such that where x = (y, z) ∈ RN with y ∈ Rm, z ∈ Rn, N = m + n ≥ 3, m ≥ 2, n ≥ 1, 1 < p < (N + 2)/(N − 2), ω ⊆ Rm a bounded C1,1 domain and Ω = ω × Rn, then the Dirichlet problem −Δu + u = b(x)|u|p−1u in Ω has a solution that changes sign in Ω, in addition to a positive solution.

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