Positive solutions of a one-dimensional indefinite capillarity-type problem: a variational approach
type: Item
instance of: scholarly article
title: Positive solutions of a one-dimensional indefinite capillarity-type problem: a variational approach
zbMATH Open document ID: 1362.34042
author: Julián López-Gómez
author: Pierpaolo Omari
author: Sabrina Rivetti
published in: Journal of Differential Equations
publication date: 2016-12-22T00:00:00Z
review text: The authors consider the quasilinear Neumann problem \[ -\Biggl(\frac{u'}{\sqrt{1+(u')^2}}\Biggr)'=\alpha(x)f(u) \text{ in }(0,1),\quad u'(0)=u'(1)=0,\tag{P} \] where \(\alpha\in L^1(0,1)\) is a sign-changing function, and \(f\in C([0,+\infty))\) is a nonnegative function which is sublinear or superlinear at \(0\) and \(+\infty\). The authors establish some existence results of bounded variation functions that are positive or strictly positive solutions to problem (P) in the following sense: Set \[ \mathcal J(v)=\sup_{_{\substack{ w_1,w_2\in C_0^1(0,1) \\ \|w_1^2+w_2^2\|_{L^\infty} \leq 1 }}}\int_0^1(vw_1+w_2)dx-1 \] for every \(v\in BV(0,1)\). A nonnegative function \(u\in BV(0,1)\) is a solution to problem (P) if the variational inequality \[ \mathcal{J}(v)-\mathcal{J}(u)\geq \int_0^1a(x)f(u(x))(v(x)-u(x))dx\text{ for every }v\in BV(0,1), \] holds. A solution \(u\) is said to be positive if \(\operatorname{ess}\sup_{(0,1)} u>0\), and strictly positive if \(\operatorname{ess}\inf_{(0,1)} u>0\). Regularity properties of the solutions are also studied. Going into detail, the authors prove that for each open interval \((\alpha,\beta)\subset (0,1)\), with \(\operatorname{ess}\inf _{(\alpha,\beta)}a\geq 0\), the found solution \(u\) belongs to \(W_{\mathrm{loc}}^{1,2}(\alpha,\beta)\cap W^{1,1}(\alpha,\beta)\). Then, in particular, \(u\) classically solves the equation in \((\alpha,\beta)\). In addiction, for intervals \((\alpha,\beta)\subset (0,1)\) such that \(\operatorname{ess}\inf_{(\alpha,\gamma)}a\geq 0\) and \(\operatorname{ess}\sup_{(\gamma,\beta)}a\leq 0\), with \(\gamma\in (\alpha,\beta)\), the types of possible discontinuities of \(u\) at \(\gamma\) are investigated. Finally, further properties of the solution \(u\), including the strict positiveness, are established in the case \(a\) changes sign only finitely many times in \((0,1)\). For the parameterized version of problem \((P)\) where the second hand-side is multiplied by a positive parameter \(\lambda\),…
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 76D45
zbMATH DE Number: 6668279
zbMATH Keywords: prescribed curvature equation
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: indefinite weight
zbMATH Keywords: bounded variation solution
zbMATH Keywords: regularization procedure
zbMATH Keywords: variational method
reviewed by: Giovanni Anello
MaRDI profile type: Publication
full work available at URL: j.jde.2016.10.046
OpenAlex ID: W2552568133
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Positive solutions of a one-dimensional indefinite capillarity-type problem: a variational approach (DOI)
DOI: 10.1016/J.JDE.2016.10.046