scholarly journals MULTIPLE SOLUTIONS FOR EQUATIONS OF p(x)-LAPLACE TYPE WITH NONLINEAR NEUMANN BOUNDARY CONDITION

2016 ◽  
Vol 53 (6) ◽  
pp. 1805-1821
Author(s):  
Yun-Ho Ki ◽  
Kisoeb Park
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Neumann Boundary ◽  
Laplace Type

Multiple Solutions for a Class of Neumann Elliptic Problems on Compact Riemannian Manifolds with Boundary

2010 ◽  
Vol 53 (4) ◽  
pp. 674-683 ◽  
Author(s):  
Alexandru Kristály ◽  
Nikolaos S. Papageorgiou ◽  
Csaba Varga
Keyword(s):  
Boundary Condition ◽  
Riemannian Manifolds ◽  
Neumann Boundary Condition ◽  
Multiple Solutions ◽  
Compact Riemannian Manifold ◽  
Neumann Boundary ◽  
Manifolds With Boundary ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Manifold With Boundary

AbstractWe study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

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.

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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