scholarly journals A Semilinear Dirichlet Problem

1979 ◽  
Vol 31 (2) ◽  
pp. 337-340 ◽  
Author(s):  
Alfonso Castro
Keyword(s):  
Dirichlet Problem ◽  
Weak Solutions ◽  
Laplacian Operator ◽  
Bounded Region ◽  
Existence Of Weak Solutions ◽  
Gradient Operator ◽  
Carathéodory Condition ◽  
Caratheodory Condition ◽  
Image Position

Introduction and notations. Let Ω be a bounded region in Rn. In this note we discuss the existence of weak solutions (see [4, Section 2]) of the Dirichlet problem(I)where Δ is the Laplacian operator, g : Ω × R → R and f : Ω × Rn+1 → R are functions satisfying the Caratheodory condition (see [2, Section 3]), and ∇ is the gradient operator.We let λ1 < λ2 ≦ … ≦ λm ≦ … denote the sequence of numbers for which the problem(II)has nontrivial weak solutions.

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).

scholarly journals Eigenvalues for a Neumann Boundary Problem Involving thep(x)-Laplacian

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Qing Miao
Keyword(s):  
Variational Principle ◽  
Boundary Problem ◽  
Neumann Problem ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Ekeland’S Variational Principle ◽  
Laplacian Operator ◽  
Ekeland's Variational Principle ◽  
Existence Of Weak Solutions

We study the existence of weak solutions to the following Neumann problem involving thep(x)-Laplacian operator:  -Δp(x)u+e(x)|u|p(x)-2u=λa(x)f(u),in  Ω,∂u/∂ν=0,on  ∂Ω. Under some appropriate conditions on the functionsp,  e,  a, and  f, we prove that there existsλ¯>0such that anyλ∈(0,λ¯)is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland’s variational principle.

Existence theorems for fractional p-Laplacian problems

2016 ◽  
Vol 15 (05) ◽  
pp. 607-640 ◽  
Author(s):  
Paolo Piersanti ◽  
Patrizia Pucci
Keyword(s):  
Weak Solutions ◽  
Existence Of Solutions ◽  
Existence Theorems ◽  
Positive Function ◽  
Existence Results ◽  
Laplacian Operator ◽  
Bounded Domains ◽  
Lipschitz Boundary ◽  
Nontrivial Solutions ◽  
Existence Of Weak Solutions

The paper focuses on the existence of nontrivial solutions of a nonlinear eigenvalue perturbed problem depending on a real parameter [Formula: see text] under homogeneous boundary conditions in bounded domains with Lipschitz boundary. The problem involves a weighted fractional [Formula: see text]-Laplacian operator. Denoting by [Formula: see text] a sequence of eigenvalues obtained via mini–max methods and linking structures we prove the existence of (weak) solutions both when there exists [Formula: see text] such that [Formula: see text] and when [Formula: see text]. The paper is divided into two parts: in the first part existence results are determined when the perturbation is the derivative of a globally positive function whereas, in the second part, the case when the perturbation is the derivative of a function that could be either locally positive or locally negative at [Formula: see text] is taken into account. In the latter case, it is necessary to extend the main results reported in [A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional [Formula: see text]-Laplacian problems via Morse theory, Adv. Calc. Var. 9(2) (2016) 101–125]. In both cases, the existence of solutions is achieved via linking methods.

scholarly journals Weak solutions for a (p(z),q(z))-Laplacian Dirichlet problem

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 999-1011
Author(s):  
Antonella Nastasi
Keyword(s):  
Dirichlet Problem ◽  
Weak Solutions ◽  
Nonnegative Solution ◽  
Laplacian Operator ◽  
Potential Term ◽  
Reaction Term ◽  
Double Phase

We establish the existence of a nontrivial and nonnegative solution for a double phase Dirichlet problem driven by a (p(z); q(z))-Laplacian operator plus a potential term. Our approach is variational, but the reaction term f need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition.

scholarly journals The Multiplicity of Nontrivial Solutions for a New p x -Kirchhoff-Type Elliptic Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Chang-Mu Chu ◽  
Yu-Xia Xiao
Keyword(s):  
Variational Principle ◽  
Elliptic Problem ◽  
Weak Solutions ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Laplacian Operator ◽  
Nontrivial Solutions ◽  
Nonlocal Problems ◽  
Existence Of Weak Solutions ◽  
Kirchhoff Problem

In the paper, we study the existence of weak solutions for a class of new nonlocal problems involving a p x -Laplacian operator. By using Ekeland’s variational principle and mountain pass theorem, we prove that the new p x -Kirchhoff problem has at least two nontrivial weak solutions.

scholarly journals Existence of Solutions for the Evolution -Laplacian Equation Not in Divergence Form

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Changchun Liu ◽  
Junchao Gao ◽  
Songzhe Lian
Keyword(s):  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximate Solutions ◽  
Divergence Form ◽  
Existence Of Weak Solutions ◽  
Uniform Estimates ◽  
Laplacian Equation

The existence of weak solutions is studied to the initial Dirichlet problem of the equation , with inf . We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions.

On the existence of weak solutions for a nonlinear time dependent Dirac equation

1989 ◽  
Vol 113 (1-2) ◽  
pp. 149-158 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Dirac Equation ◽  
Characteristic Function ◽  
Weak Solutions ◽  
Time Dependent ◽  
Existence Of Weak Solutions ◽  
Nonlinear Dirac Equation ◽  
The Cauchy Problem ◽  
Image Position

SynopsisIn this paper we prove the existence of a weak solution of the Cauchy problem for the nonlinear Dirac equation in ℝ × ℝwhere X(r) is the characteristic function of a compact interval of ]0, + ∞[

scholarly journals Quasilinear Dirichlet problems with competing operators and convection

2020 ◽  
Vol 18 (1) ◽  
pp. 1510-1517
Author(s):  
Dumitru Motreanu
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Weak Solutions ◽  
Existence Of Solutions ◽  
Approximation Procedure ◽  
Convection Term ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Variational Structure

Abstract The paper deals with a quasilinear Dirichlet problem involving a competing (p,q)-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing (p,q)-Laplacian we consider the usual (p,q)-Laplacian, our results ensure the existence of weak solutions.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

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