We prove the existence of multiple positive BV-solutions of the Neumann problem -(rac{u'}{sqrt{1+u'^2}} ight)'=a(x)f(u) in (0,1), u'(0)=u'(1)=0, where $a(x) > 0$ and $f$ belongs to a class of nonlinear functions whose prototype example is given by $f(u) = -lambda u + u^p$, for $lambda > 0$ and $p > 1$. In particular, $f(0)=0$ and $f$ has a unique positive zero, denoted by $u_0$. Solutions are distinguished by the number of intersections (in a generalized sense) with the constant solution $u = u_0$. We further prove that the solutions found have continuous energy and we also give sufficient conditions on the nonlinearity to get classical solutions. The analysis is performed using an approximation of the mean curvature operator and the shooting method.
Alberto Boscaggin, Francesca Colasuonno, Colette De Coster (2021). Multiple bounded variation solutions for a prescribed mean curvature equation with Neumann boundary conditions. JOURNAL OF DIFFERENTIAL EQUATIONS, 285, 607-639 [10.1016/j.jde.2021.03.021].
Multiple bounded variation solutions for a prescribed mean curvature equation with Neumann boundary conditions
Francesca Colasuonno
;
2021
Abstract
We prove the existence of multiple positive BV-solutions of the Neumann problem -(rac{u'}{sqrt{1+u'^2}} ight)'=a(x)f(u) in (0,1), u'(0)=u'(1)=0, where $a(x) > 0$ and $f$ belongs to a class of nonlinear functions whose prototype example is given by $f(u) = -lambda u + u^p$, for $lambda > 0$ and $p > 1$. In particular, $f(0)=0$ and $f$ has a unique positive zero, denoted by $u_0$. Solutions are distinguished by the number of intersections (in a generalized sense) with the constant solution $u = u_0$. We further prove that the solutions found have continuous energy and we also give sufficient conditions on the nonlinearity to get classical solutions. The analysis is performed using an approximation of the mean curvature operator and the shooting method.File | Dimensione | Formato | |
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