Global bifurcation for problem with mean curvature operator on general domain

2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Guowei Dai
Keyword(s):  
Mean Curvature ◽  
Global Bifurcation ◽  
Curvature Operator ◽  
General Domain ◽  
Mean Curvature Operator

scholarly journals Nodal Solutions for Problems with Mean Curvature Operator in Minkowski Space with Nonlinearity Jumping Only at the Origin

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Wenguo Shen
Keyword(s):  
Minkowski Space ◽  
Mean Curvature ◽  
Global Bifurcation ◽  
Connected Components ◽  
Curvature Operator ◽  
Linear Perturbation ◽  
Open Ball ◽  
Nodal Solutions ◽  
Mean Curvature Operator ◽  
Perturbation Problems

In this paper, we establish a unilateral global bifurcation result for half-linear perturbation problems with mean curvature operator in Minkowski space. As applications of the abovementioned result, we shall prove the existence of nodal solutions for the following problem −div∇v/1−∇v2=αxv++βxv−+λaxfv, in BR0,vx=0, on ∂BR0, where λ ≠ 0 is a parameter, R is a positive constant, and BR0=x∈ℝN:x<R is the standard open ball in the Euclidean space ℝNN≥1 which is centered at the origin and has radius R. a(|x|) ∈ C[0, R] is positive, v+ = max{v, 0}, v− = −min{v, 0}, α(|x|), β(|x|) ∈ C[0, R]; f∈Cℝ,ℝ, s f (s) > 0 for s ≠ 0, and f0 ∈ [0, ∞], where f0 = lim|s|⟶0 f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

scholarly journals Positive solutions of the discrete Dirichlet problem involving the mean curvature operator

2019 ◽  
Vol 17 (1) ◽  
pp. 1055-1064 ◽  
Author(s):  
Jiaoxiu Ling ◽  
Zhan Zhou
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Mean Curvature ◽  
Nonlinear Term ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Curvature Operator ◽  
Mean Curvature Operator ◽  
The Mean

Abstract In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

Nontrivial Solutions for Potential Systems Involving the Mean Curvature Operator in Minkowski Space

2017 ◽  
Vol 17 (4) ◽  
pp. 769-780 ◽  
Author(s):  
Daniela Gurban ◽  
Petru Jebelean ◽  
Călin Şerban
Keyword(s):  
Minkowski Space ◽  
Critical Point Theory ◽  
Mean Curvature ◽  
Lower Semicontinuous ◽  
Point Theory ◽  
Curvature Operator ◽  
Nontrivial Solutions ◽  
Potential System ◽  
Mean Curvature Operator ◽  
The Mean

AbstractIn this paper, we use the critical point theory for convex, lower semicontinuous perturbations of{C^{1}}-functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator{u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})}. The solvability of a general non-potential system is also established.

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