scholarly journals A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition

2017 ◽  
Vol 37 (3) ◽  
pp. 1691-1706 ◽  
Author(s):  
Khadijah Sharaf ◽  
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Perturbation Result

Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions

2014 ◽  
Vol 16 (04) ◽  
pp. 1350048
Author(s):  
Shuangjie Peng ◽  
Yanfang Peng
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Variational Method ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Singular Elliptic Equation ◽  
Sign Changing Solutions ◽  
Least Energy

We study the following singular elliptic equation [Formula: see text] with Dirichlet boundary condition, which is related to the well-known Caffarelli–Kohn–Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.

Non-stabilizing solutions of semilinear hyperbolic and elliptic equations with damping

2003 ◽  
Vol 133 (5) ◽  
pp. 1137-1153 ◽  
Author(s):  
M. A. Jendoubi ◽  
P. Poláčik
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Elliptic Equation ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Cylindrical Domain ◽  
Damped Wave Equation ◽  
Bounded Solutions ◽  
Single Function

We consider two types of equations on a cylindrical domain Ω × (0, ∞), where Ω is a bounded domain in RN, N ≥ 2. The first type is a semilinear damped wave equation, in which the unbounded direction of Ω × (0, ∞) is reserved for time t. The second type is an elliptic equation with a singled-out unbounded variable t. In both cases, we consider solutions that are defined and bounded on Ω × (0, ∞) and satisfy a Dirichlet boundary condition on ∂Ω × (0, ∞). We show that, for some nonlinearities, the equations have bounded solutions that do not stabilize to any single function φ: Ω → R, as t → ∞; rather, they approach a continuum of such functions. This happens despite the presence of damping in the equation that forces the t derivative of bounded solutions to converge to 0 as t → ∞. Our results contrast with known stabilization properties of solutions of such equations in the case N = 1.

scholarly journals On the extremal solutions of semilinear elliptic problems

2005 ◽  
Vol 2005 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Lamia Ben Chaabane
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Semilinear Elliptic Equation ◽  
Extremal Solutions ◽  
Bounded Smooth Domain ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems ◽  
Bounded Smooth

We investigate here the properties of extremal solutions for semilinear elliptic equation−Δu=λf(u)posed on a bounded smooth domain ofℝnwith Dirichlet boundary condition and withfexploding at a finite positive valuea.

scholarly journals The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Giovany M. Figueiredo ◽  
A. Razani
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Elliptic Equation ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Logistic Equation ◽  
Positive Weak Solution

AbstractIn this paper, a nonhomogeneous elliptic equation of the form $$\begin{aligned}& - \mathcal{A}\bigl(x, \vert u \vert _{L^{r(x)}}\bigr) \operatorname{div}\bigl(a\bigl( \vert \nabla u \vert ^{p(x)}\bigr) \vert \nabla u \vert ^{p(x)-2} \nabla u\bigr) \\& \quad =f(x, u) \vert \nabla u \vert ^{\alpha (x)}_{L^{q(x)}}+g(x, u) \vert \nabla u \vert ^{ \gamma (x)}_{L^{s(x)}} \end{aligned}$$ − A ( x , | u | L r ( x ) ) div ( a ( | ∇ u | p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) | ∇ u | L q ( x ) α ( x ) + g ( x , u ) | ∇ u | L s ( x ) γ ( x ) on a bounded domain Ω in ${\mathbb{R}}^{N}$ R N ($N >1$ N > 1 ) with $C^{2}$ C 2 boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

scholarly journals Gradient estimates for weighted harmonic function with Dirichlet boundary condition

2021 ◽  
Vol 213 ◽  
pp. 112498
Author(s):  
Nguyen Thac Dung ◽  
Jia-Yong Wu
Keyword(s):  
Boundary Condition ◽  
Harmonic Function ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Gradient Estimates

On the Dispersive Estimate for the Dirichlet Schrödinger Propagator and Applications to Energy Critical NLS

2014 ◽  
Vol 66 (5) ◽  
pp. 1110-1142
Author(s):  
Dong Li ◽  
Guixiang Xu ◽  
Xiaoyi Zhang
Keyword(s):  
Boundary Condition ◽  
Fundamental Solution ◽  
Unit Ball ◽  
Dirichlet Boundary Condition ◽  
Obstacle Problem ◽  
Dirichlet Boundary ◽  
Radial Symmetry ◽  
Well Posedness ◽  
Robust Algorithm ◽  
Dispersive Estimate

AbstractWe consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator and give a robust algorithm to prove sharp L1 → L∞ dispersive estimates. We showcase the analysis in dimensions n = 5, 7. As an application, we obtain global well–posedness and scattering for defocusing energy-critical NLS on with Dirichlet boundary condition and radial data in these dimensions.

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