scholarly journals Multiplicity Results for thepx-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
K. Saoudi ◽  
M. Kratou ◽  
S. Alsadhan
Keyword(s):  
Boundary Condition ◽  
Neumann Problem ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Neumann Boundary ◽  
Positive Parameter ◽  
Multiplicity Results ◽  
Laplacian Equation ◽  
Variational Techniques ◽  
Singular Nonlinearities

We investigate the singular Neumann problem involving thep(x)-Laplace operator:Pλ{-Δpxu+|u|px-2u  =1/uδx+fx,u, in  Ω;  u>0,  in  Ω;  ∇upx-2∂u/∂ν=λuqx,  on  ∂Ω}, whereΩ⊂RNN≥2is a bounded domain withC2boundary,λis a positive parameter, andpx,qx,δx, andfx,uare assumed to satisfy assumptions(H0)–(H5)in the Introduction. Using some variational techniques, we show the existence of a numberΛ∈0,∞such that problemPλhas two solutions forλ∈0,Λ,one solution forλ=Λ, and no solutions forλ>Λ.

BLOW-UP ANALYSIS FOR SOLUTIONS OF THE LAPLACIAN EQUATION WITH EXPONENTIAL NEUMANN BOUNDARY CONDITION IN DIMENSION TWO

2006 ◽  
Vol 08 (06) ◽  
pp. 737-761 ◽  
Author(s):  
YU-XIA GUO ◽  
JIA-QUAN LIU
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Existence Theorem ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Neumann Boundary ◽  
Asymptotic Behavior Of Solutions ◽  
Laplacian Equation ◽  
Blow Up Analysis

We consider the asymptotic behavior of solutions of the Laplacian equation with exponential Neumann boundary condition in dimension two. As an application, we prove an existence theorem of nonminimum solutions.

scholarly journals EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A NEUMANN PROBLEM INVOLVING VARIABLE EXPONENT GROWTH CONDITIONS

2008 ◽  
Vol 50 (3) ◽  
pp. 565-574 ◽  
Author(s):  
MARIA-MAGDALENA BOUREANU ◽  
MIHAI MIHĂILESCU
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Neumann Problem ◽  
Neumann Boundary Condition ◽  
Variable Exponent ◽  
Main Tool ◽  
Neumann Boundary ◽  
Growth Conditions ◽  
Linear Elliptic Equation ◽  
Existence And Multiplicity

AbstractIn this paper we study a non-linear elliptic equation involving p(x)-growth conditions and satisfying a Neumann boundary condition on a bounded domain. For that equation we establish the existence of two solutions using as a main tool an abstract linking argument due to Brézis and Nirenberg.

An Ambrosetti–Prodi-type result for a quasilinear Neumann problem

2012 ◽  
Vol 55 (3) ◽  
pp. 771-780 ◽  
Author(s):  
Franciso Odair de Paiva ◽  
Marcelo Montenegro
Keyword(s):  
Boundary Condition ◽  
Neumann Problem ◽  
Neumann Boundary Condition ◽  
A Priori ◽  
Degree Theory ◽  
Neumann Boundary ◽  
Minimal Solution ◽  
Type Result

AbstractWe study the problem −∆pu = f(x, u) + t in Ω with Neumann boundary condition |∇u|p−2(∂u/∂v) = 0 on ∂Ω. There exists a t0 ∈ ℝ such that for t > t0 there is no solution. If t ≤ t0, there is at least a minimal solution, and for t < t0 there are at least two distinct solutions. We use the sub–supersolution method, a priori estimates and degree theory.

scholarly journals $W^{1,N}$ versus $C^1$ local minimizer for a singular functional with Neumann boundary condition

2017 ◽  
Vol 37 (1) ◽  
pp. 71
Author(s):  
Kamel Saoudi
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Local Minimum ◽  
Smooth Boundary ◽  
Lagrange Equation ◽  
Neumann Boundary ◽  
Local Minimizer ◽  
Multiplicity Of Positive Solutions

Let $\Omega\subset\R^N,$ be a bounded domain with smooth boundary. Let $g:\R^+\to\R^+$ be a continuous on $(0,+\infty)$ non-increasing and satisfying $$c_1=\liminf_{t\to 0^+}g(t)t^{\delta}\leq\underset{t\to 0^+}{\limsup} g(t)t^{\delta}=c_2,$$ for some $c_1,c_2>0$ and $0<\delta<1.$ Let $f(x,s) = h(x,s)e^{bs^{\frac{N}{N-1}}},$ $b>0$ is a constant.Consider the singular functional $I: W^{1,N}(\Omega)\to \R$ defined as \begin{eqnarray*}&&I(u)\eqdef\frac{1}{N}\|u\|^N_{W^{1,N}(\Omega)}-\int_{\Omega}G(u^+)\,{\rm d} x-\int_{\Omega}F(x,u^+) \,{\rm d} x\nonumber\\&& -\frac{1}{q+1}||u||^{q+1}_{L^{q+1}(\partial\Omega)}\nonumber\end{eqnarray*} where $F(x,u)=\int_0^sf(x,s)\,{\rm d}s$, $G(u)=\int_0^s g(s)\,{\rm d}s$. We show that if $u_0\in C^1(\overline{\Omega})$ satisfying $u_0\geq \eta \mbox{dist}(x,\partial\Omega)$, for some $0<\eta$, is a local minimum of $I$ in the $C^1(\overline{\Omega})\cap C_0(\overline{\Omega})$ topology, then it is also a local minimum in $W^{1,N}(\Omega)$ topology. This result is useful %for proving multiple solutions to the associated Euler-lagrange equation ${\rm (P)}$ defined below.to prove the multiplicity of positive solutions to critical growth problems with co-normalboundary conditions.

scholarly journals Deriving The Upper Blow-up Rate Estimate for a Parabolic Problem

2020 ◽  
pp. 200-203
Author(s):  
Maan A. Rasheed
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Parabolic Problem ◽  
Blow Up ◽  
Neumann Boundary ◽  
Rate Estimate ◽  
Blow Up Rate

In this paper, the blow-up solutions for a parabolic problem, defined in a bounded domain, are studied. Namely, we consider the upper blow-up rate estimate for heat equation with a nonlinear Neumann boundary condition defined on a ball in Rn.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

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