Infinitely Many Solutions for Semilinear Strongly Degenerate Elliptic Differential Equations with Lack of Symmetry

2021 ◽  
Vol 175 (1) ◽  
Author(s):  
Duong Trong Luyen
Keyword(s):  
Differential Equations ◽  
Infinitely Many Solutions ◽  
Elliptic Differential Equations ◽  
Degenerate Elliptic ◽  
Strongly Degenerate

scholarly journals Hölder continuity of solutions of some degenerate elliptic differential equations

2000 ◽  
Vol 62 (3) ◽  
pp. 369-377 ◽  
Author(s):  
Ahmed Mohammed
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Weak Solutions ◽  
Function Spaces ◽  
Elliptic Differential Equation ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Elliptic Differential Equations ◽  
Continuity Of Solutions ◽  
Degenerate Elliptic

Weak solutions of the degenerate elliptic differential equation Lu := −div(A (x)∇u)+b·∇u+Vu = f, with |b|2ω−1, V, f in some appropriate function spaces, will be shown to be Hölder continuous.

Infinitely many solutions for semilinear Δλ-Laplace equations with sign-changing potential and nonlinearity

2017 ◽  
Vol 54 (4) ◽  
pp. 536-549 ◽  
Author(s):  
Jianhua Chen ◽  
Xianhua Tang ◽  
Zu Gao
Keyword(s):  
Elliptic Operator ◽  
Bounded Domain ◽  
Boundary Value ◽  
Quadratic Growth ◽  
Infinitely Many Solutions ◽  
Type Condition ◽  
Degenerate Elliptic Operator ◽  
Laplace Equations ◽  
Degenerate Elliptic ◽  
Strongly Degenerate

In this paper, we prove the existence of infinitely many solutions for the following class of boundary value elliptic problems where Ω is a bounded domain in RN (N ≥ 2), Δλ is a strongly degenerate elliptic operator, V (x) is allowing to be sign-changing and f is a function with a more general super-quadratic growth, which is weaker than the Ambrosetti-Rabinowitz type condition.

Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system

2017 ◽  
Vol 8 (1) ◽  
pp. 661-678 ◽  
Author(s):  
Cung The Anh ◽  
Bui Kim My
Keyword(s):  
Elliptic System ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Smooth Boundary ◽  
Infinitely Many Solutions ◽  
Degenerate Elliptic System ◽  
Degenerate Elliptic ◽  
Degenerate Operator ◽  
Strongly Degenerate

Abstract We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system \left\{\begin{aligned} &\displaystyle{-}\Delta_{\lambda}u=\lvert v\rvert^{p-1}% v&&\displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle{-}\Delta_{\lambda}v=\lvert u\rvert^{q-1}u&&\displaystyle% \phantom{}\text{in }\Omega,\\ &\displaystyle u=v=0&&\displaystyle\phantom{}\text{on }\partial\Omega,\end{% aligned}\right. in a bounded domain {\Omega\subset\mathbb{R}^{N}} with smooth boundary {\partial\Omega} . Here {p,q>1} , and {\Delta_{\lambda}} is the strongly degenerate operator of the form \Delta_{\lambda}u=\sum^{N}_{j=1}\frac{\partial}{\partial x_{j}}\Bigl{(}\lambda% _{j}^{2}(x)\frac{\partial u}{\partial x_{j}}\Bigr{)}, where {\lambda(x)=(\lambda_{1}(x),\dots,\lambda_{N}(x))} satisfies certain conditions.

scholarly journals Weak solutions of a quasi-linear degenerate elliptic system with discontinuous coefficients

1984 ◽  
Vol 94 ◽  
pp. 105-135
Author(s):  
Yoshiaki Ikeda
Keyword(s):  
Differential Equations ◽  
Elliptic System ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Discontinuous Coefficients ◽  
Degenerate Elliptic System ◽  
Elliptic Differential Equations ◽  
Degenerate Elliptic ◽  
Linear Degenerate

We shall discuss regularities and related topics on weak solutions of the system of the following quasi-linear elliptic differential equations (a combination of almost single equations)in a bounded domain Ω in Rn (n ≧ 2), where A1j … (A1j …, Anj) are given vector functions of (x, u, ▽uj), Bj are scalar functions of the same variables, and ▽uj = (∂uj/∂x1, …, ∂uj/∂xj denote the gradients of the uj = uj(x) (j = 1, …, m).

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