Positive solutions of a one-dimensional indefinite capillarity-type problem: a variational approach
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 \(\lambda\), the authors prove the existence of at least two positive solutions for all \(\lambda\) sufficiently small or sufficiently large according to the type of behavior (sublinear or superlinear) of \(f\) at \(0\) and \(+\infty\). The proofs of the existence results are based on an approximation procedure: the authors construct a sequence of approximating problems whose associated energy functional is of class \(C^1\) in \(W^{1,\rho}(0,1)\), for some convenient \(\rho\in (0,1)\). Then, a solution to each approximating problem if found out by minimax or minimization methods, according to the type of behavior of \(f\) at \(0\) and \(+\infty\). Finally, using appropriate uniform estimates in \(W^{1,1}(0,1)\), it is proved that the sequence of solutions of the approximating problems is uniformly bounded and strongly converges, in \(L^1(0,1)\), to a solution of problem (P).
scientific article; zbMATH DE number 6668279
scientific article