Nodal solutions to problem with mean curvature operator in Minkowski space. (English) Zbl 1424.35187

Dai, Guowei; Wang, Jun

Differ. Integral Equ. 30, No. 5-6, 463-480 (2017).

Summary: This paper is devoted to investigate the existence and multiplicity of radial nodal solutions for the following Dirichlet problem with mean curvature operator in Minkowski space \[ \begin{cases} -\operatorname{div}\Big (\frac{\nabla v}{\sqrt{1-| \nabla v|^2}} \Big)=\lambda f(| x|,v)\,\, &\text{in}\,\, B_R(0),\\ v=0&\text{on}\,\,\partial B_R(0).\end{cases} \] By bifurcation approach, we determine the interval of parameter \(\lambda\) in which the above problem has two or four radial nodal solutions which have exactly \(n-1\) simple zeros in \((0,R)\) according to linear/sublinear/superlinear nonlinearity at zero. The asymptotic behaviors of radial nodal solutions as \(\lambda \to +\infty\) and \(n\to +\infty\) are also studied.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
34C23	Bifurcation theory for ordinary differential equations
35B40	Asymptotic behavior of solutions to PDEs
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

unbounded component; superlinear growth; sublinear growth; compactness