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Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefinite weight. (English) Zbl 1443.34024

The paper deals with the existence of positive radial solutions of a boundary value problem of the form \[ \operatorname{div}(\nabla{u}/\sqrt{1-|\nabla u|^2})+\lambda a(|x|)u^p=0,\tag{\(\ast\)} \] in \(B\) and \(\partial_{\nu}u=0\) on \(\partial B,\) where \(B\) is a ball in the \(N\)-dimensional Euclidean space, centered at the origin. Actually the problem is transformed into \[ (r^{N-1}u'(1-(u'^2))^{-1/2})'+\lambda r^{N-1}a(r)g(u)=0,\quad u'(0)=u'(R)=0,\tag{\(\ast\ast\)} \] where \(R\) is the radius of the ball \(B.\) The authors provide sufficient conditions to guarantee the fact that there is some \(\lambda^*>0\) such that, for every \(\lambda>\lambda^*\), there exist at least two positive solutions of problem \(({**}),\) which results that there are at least two positive solutions of the problem \((*)\). The idea of the proof is to use the Leray-Schauder degree theory on the parameter auxiliary boundary value problem \[ (r^{N-1}u'(1-(u'^2))^{-1/2})'+\vartheta r^{N-1}[f(r,u)+\alpha v(r)]=0, \;u'(0)=u'(R)=0, \] where \(f(r,u)=\lambda a(r)g(u)\) if \(u\geq 0\) and \(=-u\) if \(u<0\).

MSC:

34B09 Boundary eigenvalue problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
35B09 Positive solutions to PDEs
35J15 Second-order elliptic equations
47H11 Degree theory for nonlinear operators
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References:

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