Existence of solution for quasilinear equations involving local conditions

2019 ◽  
Vol 150 (6) ◽  
pp. 3074-3086
Author(s):  
Patricio Cerda ◽  
Leonelo Iturriaga
Keyword(s):  
Continuous Function ◽  
Nonlinear Term ◽  
Quasilinear Equation ◽  
Existence Of Solution ◽  
Positive Parameter ◽  
Quasilinear Equations ◽  
Local Conditions ◽  
Existence Of Weak Solutions ◽  
Carathéodory Function ◽  
Image Position

AbstractIn this paper, we study the existence of weak solutions of the quasilinear equation \begin{cases} -{\rm div} (a(\vert \nabla u \vert ^2)\nabla u)=\lambda f(x,u) &{\rm in} \ \Omega,\\ u=0 &{\rm on} \ \partial\Omega, \end{cases}where a : ℝ → [0, ∞) is C1 and a nonincreasing continuous function near the origin, the nonlinear term f : Ω × ℝ → ℝ is a Carathéodory function verifying certain superlinear conditions only at zero, and λ is a positive parameter. The existence of the solution relies on C1-estimates and variational arguments.

scholarly journals A MULTIPLICITY RESULT FOR A CLASS OF EQUATIONS OFp-LAPLACIAN TYPE WITH SIGN-CHANGING NONLINEARITIES

2009 ◽  
Vol 51 (3) ◽  
pp. 513-524 ◽  
Author(s):  
NGUYEN THANH CHUNG ◽  
QUỐC ANH NGÔ
Keyword(s):  
Bounded Domain ◽  
Multiplicity Result ◽  
Positive Parameter ◽  
Existence And Multiplicity ◽  
Carathéodory Function ◽  
Image Position

AbstractUsing variational arguments we study the non-existence and multiplicity of non-negative solutions for a class equations of the formwhere Ω is a bounded domain inN,N≧ 3,fis a sign-changing Carathéodory function on Ω × [0, +∞) and λ is a positive parameter.

A result of multiplicity of solutions for a class of quasilinear equations

2012 ◽  
Vol 55 (2) ◽  
pp. 291-309 ◽  
Author(s):  
Claudianor O. Alves ◽  
Giovany M. Figueiredo ◽  
Uberlandio B. Severo
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Category Theory ◽  
Nonlinear Term ◽  
Positive Parameter ◽  
Quasilinear Equations ◽  
Subcritical Growth ◽  
Minimax Methods ◽  
Multiplicity Of Positive Solutions

AbstractWe establish the multiplicity of positive weak solutions for the quasilinear Dirichlet problem−Lpu+ |u|p−2u=h(u)in Ωλ,u= 0 on ∂Ωλ, where Ωλ= λΩ, Ω is a bounded domain in ℝN, λ is a positive parameter,Lpu≐ Δpu+ Δp(u2)uand the nonlinear termh(u) has subcritical growth. We use minimax methods together with the Lyusternik–Schnirelmann category theory to get multiplicity of positive solutions.

scholarly journals Multiplicity of Small or Large Energy Solutions for Kirchhoff–Schrödinger-Type Equations Involving the Fractional p-Laplacian in RN

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 436 ◽  
Author(s):  
Jae-Myoung Kim ◽  
Yun-Ho Kim ◽  
Jongrak Lee
Keyword(s):  
Continuous Function ◽  
Variational Methods ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Term ◽  
The Other ◽  
Large Energy ◽  
Small Energy ◽  
Multiplicity Results ◽  
Carathéodory Function

We herein discuss the following elliptic equations: M ∫ R N ∫ R N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y ( - Δ ) p s u + V ( x ) | u | p - 2 u = λ f ( x , u ) i n R N , where ( - Δ ) p s is the fractional p-Laplacian defined by ( - Δ ) p s u ( x ) = 2 lim ε ↘ 0 ∫ R N ∖ B ε ( x ) | u ( x ) - u ( y ) | p - 2 ( u ( x ) - u ( y ) ) | x - y | N + p s d y , x ∈ R N . Here, B ε ( x ) : = { y ∈ R N : | x - y | < ε } , V : R N → ( 0 , ∞ ) is a continuous function and f : R N × R → R is the Carathéodory function. Furthermore, M : R 0 + → R + is a Kirchhoff-type function. This study has two aims. One is to study the existence of infinitely many large energy solutions for the above problem via the variational methods. In addition, a major point is to obtain the multiplicity results of the weak solutions for our problem under various assumptions on the Kirchhoff function M and the nonlinear term f. The other is to prove the existence of small energy solutions for our problem, in that the sequence of solutions converges to 0 in the L ∞ -norm.

On a class of nonlinear anisotropic parabolic problems

2015 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
M. H. Abdou ◽  
M. Chrif ◽  
S. El Manouni ◽  
H. Hjiaj
Keyword(s):  
Sobolev Space ◽  
Parabolic Problem ◽  
Nonlinear Term ◽  
Parabolic Problems ◽  
Anisotropic Sobolev Space ◽  
Existence Of Weak Solutions ◽  
Strongly Nonlinear ◽  
Nonlinear Parabolic ◽  
Sign Conditions ◽  
Image Position

We prove the existence of weak solutions for the strongly nonlinear parabolic problem in the anisotropic Sobolev space , where the data f are assumed to be in the dual, and the nonlinear term g(x, t, s) has growth and sign conditions on s.

scholarly journals Existence results for first derivative dependent ϕ-Laplacian boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imran Talib ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuous Functions ◽  
Existence Of Solution ◽  
Existence Results ◽  
Main Concern ◽  
First Derivative ◽  
Boundary Functions ◽  
Carathéodory Function

Abstract Our main concern in this article is to investigate the existence of solution for the boundary-value problem $$\begin{aligned}& (\phi \bigl(x'(t)\bigr)'=g_{1} \bigl(t,x(t),x'(t)\bigr),\quad \forall t\in [0,1], \\& \Upsilon _{1}\bigl(x(0),x(1),x'(0)\bigr)=0, \\& \Upsilon _{2}\bigl(x(0),x(1),x'(1)\bigr)=0, \end{aligned}$$ ( ϕ ( x ′ ( t ) ) ′ = g 1 ( t , x ( t ) , x ′ ( t ) ) , ∀ t ∈ [ 0 , 1 ] , ϒ 1 ( x ( 0 ) , x ( 1 ) , x ′ ( 0 ) ) = 0 , ϒ 2 ( x ( 0 ) , x ( 1 ) , x ′ ( 1 ) ) = 0 , where $g_{1}:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ g 1 : [ 0 , 1 ] × R 2 → R is an $L^{1}$ L 1 -Carathéodory function, $\Upsilon _{i}:\mathbb{R}^{3}\rightarrow \mathbb{R} $ ϒ i : R 3 → R are continuous functions, $i=1,2$ i = 1 , 2 , and $\phi :(-a,a)\rightarrow \mathbb{R}$ ϕ : ( − a , a ) → R is an increasing homeomorphism such that $\phi (0)=0$ ϕ ( 0 ) = 0 , for $0< a< \infty $ 0 < a < ∞ . We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.

scholarly journals LINEAR SYSTEMS ON IRREGULAR VARIETIES

2019 ◽  
Vol 19 (6) ◽  
pp. 2087-2125 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Continuous Function ◽  
Line Bundle ◽  
Linear Systems ◽  
Projective Variety ◽  
Geometrical Properties ◽  
Complex Projective Variety ◽  
Wide Range ◽  
Image Position ◽  
Irregular Varieties ◽  
Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

On the Cauchy problem associated with the Brinkman flow in

2021 ◽  
pp. 1-20
Author(s):  
Michel Molina Del Sol ◽  
Eduardo Arbieto Alarcon ◽  
Rafael José Iorio
Keyword(s):  
Cauchy Problem ◽  
Porous Media ◽  
Fluid Flow ◽  
Comparison Principle ◽  
Well Posedness ◽  
Quasilinear Equations ◽  
Brinkman Equations ◽  
The Cauchy Problem ◽  
Model Fluid ◽  
Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

1999 ◽  
Vol 129 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Vesa Mustonen ◽  
Matti Tienari
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Minimization Problem ◽  
Lagrange Equation ◽  
Image Position ◽  
Generalized Laplacian

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problemwhere Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equationfor some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

Homogenization of a class of quasilinear elliptic equations in high-contrast fissured media

2006 ◽  
Vol 136 (6) ◽  
pp. 1131-1155 ◽  
Author(s):  
B. Amaziane ◽  
L. Pankratov ◽  
A. Piatnitski
Keyword(s):  
Elliptic Equation ◽  
Asymptotic Behaviour ◽  
Elliptic Equations ◽  
Quasilinear Elliptic Equation ◽  
Quasilinear Equation ◽  
Quasilinear Elliptic Equations ◽  
High Contrast ◽  
Small Measure ◽  
Image Position ◽  
Quasilinear Elliptic

The aim of the paper is to study the asymptotic behaviour of the solution of a quasilinear elliptic equation of the form with a high-contrast discontinuous coefficient aε(x), where ε is the parameter characterizing the scale of the microstucture. The coefficient aε(x) is assumed to degenerate everywhere in the domain Ω except in a thin connected microstructure of asymptotically small measure. It is shown that the asymptotical behaviour of the solution uε as ε → 0 is described by a homogenized quasilinear equation with the coefficients calculated by local energetic characteristics of the domain Ω.

SIGN-CHANGING SOLUTIONS FOR AN ASYMPTOTICALLY p-LINEAR p-LAPLACIAN EQUATION IN ℝN

2013 ◽  
Vol 15 (01) ◽  
pp. 1250046 ◽  
Author(s):  
XIANGQING LIU ◽  
YUXIA GUO
Keyword(s):  
Nonlinear Term ◽  
Positive Parameter ◽  
Laplacian Equation ◽  
Sign Changing Solutions

In this paper, we study the existence of sign-changing solutions for the p-Laplacian equation [Formula: see text] where λ is a positive parameter and the nonlinear term f is superlinear at zero and asymptotically p-linear at infinity.

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