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 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.

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.

A priori estimates for superlinear and subcritical elliptic equations: the Neumann boundary condition case

2011 ◽  
Vol 137 (3-4) ◽  
pp. 525-544 ◽  
Author(s):  
Abdellaziz Harrabi ◽  
Mohameden Ould Ahmedou ◽  
Salem Rebhi ◽  
Abdelbaki Selmi
Keyword(s):  
Boundary Condition ◽  
Elliptic Equations ◽  
Neumann Boundary Condition ◽  
A Priori Estimates ◽  
A Priori ◽  
Neumann Boundary ◽  
Priori Estimates

scholarly journals Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhenghuan Gao ◽  
Peihe Wang
Keyword(s):  
Boundary Condition ◽  
Parabolic Equations ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Neumann Boundary ◽  
Smooth Solutions ◽  
Smooth Bounded Domain ◽  
Priori Estimates

<p style='text-indent:20px;'>In this paper, we establish global <inline-formula><tex-math id="M1">\begin{document}$ C^2 $\end{document}</tex-math></inline-formula> a priori estimates for solutions to the uniformly parabolic equations with Neumann boundary condition on the smooth bounded domain in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula> by a blow-up argument. As a corollary, we obtain that the solutions converge to ones which move by translation. This generalizes the viscosity results derived before by Da Lio.</p>

Inverse spectral problem of an anharmonic oscillator on a half-axis with the Neumann boundary condition

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Agil K. Khanmamedov ◽  
Nigar F. Gafarova
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Inverse Spectral Problem ◽  
Anharmonic Oscillator ◽  
Neumann Boundary ◽  
Effective Algorithm ◽  
Inverse Spectral Problems ◽  
Main Equation ◽  
Inverse Spectral ◽  
Half Axis

AbstractAn anharmonic oscillator {T(q)=-\frac{d^{2}}{dx^{2}}+x^{2}+q(x)} on the half-axis {0\leq x<\infty} with the Neumann boundary condition is considered. By means of transformation operators, the direct and inverse spectral problems are studied. We obtain the main integral equations of the inverse problem and prove that the main equation is uniquely solvable. An effective algorithm for reconstruction of perturbed potential is indicated.

scholarly journals Blow-Up Analysis for a Quasilinear Parabolic Equation with Inner Absorption and Nonlinear Neumann Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhong Bo Fang ◽  
Yan Chai
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Neumann Boundary Condition ◽  
Blow Up ◽  
Quasilinear Parabolic Equation ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Blow Up Analysis ◽  
Inner Absorption ◽  
Quasilinear Parabolic

We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee thatu(x,t)exists globally or blows up at some finite timet*. Moreover, an upper bound fort*is derived. Under somewhat more restrictive conditions, a lower bound fort*is also obtained.

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