Sturm-Liouville equations involving discontinuous nonlinearities (Q2789200)

The main result of this paper describes a well-precise interval of parameters for which there exists at least one nontrivial solution for the following Sturm-Liouville-type problem NEWLINE\[NEWLINE-(pu')'+ qu=\lambda f(x,u) \quad \mathrm{in} \, ]a, b[,\quad u(a)=u(b)=0,\tag{1}NEWLINE\]NEWLINE where \(p\), \(q \in L^\infty([a, b])\) with \(\mathrm{essinf}_{[a,b]} p > 0\) , \(\mathrm{essinf}_{[a,b]} q\geq 0\), \(\lambda>0\) is a parameter and \(f : [a, b] \times \mathbb{R} \rightarrow \mathbb{R}\) is a locally essentially bounded function such that \( x \rightarrow f(x, t)\) is measurable for every \(t \in \mathbb{R}\), there exists a set \(A \subset [a, b]\) with \(m(A)=0\) such that the set NEWLINE\[NEWLINE\mathrm{D}_f := \bigcup_{\{x \in [a,b]\setminus A\}} \{t \in {\mathbb{R}}: f(x, \cdot)\, \mathrm{is \, discontinuous\, at}\, t\} NEWLINE\]NEWLINE has measure zero, and the functions NEWLINE\[NEWLINE f^-(x, t) := \lim_{\delta \to 0^+} \mathrm{essinf}_{\{|t-z|<\delta\}} f(x, z), \quad f^+(x, t) := \lim_{\delta \to 0^+} \mathrm{esssup}_{\{|t-z|<\delta\}} f(x, z), NEWLINE\]NEWLINE are superpositionally measurable, that is, \(f^-(x, u(x))\) and \(f^+(x, u(x))\) are measurable for all measurable functions \(u : [a, b] \to {\mathbb{R}}\).NEWLINENEWLINEAmong others, an interesting consequence of such result is the followingNEWLINENEWLINETheorem. Let \(f : {\mathbb{R}}\to {\mathbb{R}}\) be a locally essentially bounded and almost everywhere continuous function satisfying \(\inf_{{\mathbb{R}}} f > 0\). Then there exists a number \(\overline{\lambda}>0\) such that, for each \(\lambda \in ]0, \overline{\lambda}[\), the problem NEWLINE\[NEWLINE-u'' = \lambda f(u), \quad \mathrm{in} \, ]a, b[,\quad u(a)=u(b)=0, NEWLINE\]NEWLINE admits at least one nontrivial positive solution.NEWLINENEWLINETo apply the variational methods to study problem (1), the authors give a new version of an abstract nonsmooth critical point theorem which guarantees the existence of at least one nontrivial local minimum for functionals of type \(I_\lambda=\Phi-\lambda\Psi\) with \(\Phi\) and \(\Psi\) defined on a real Banach space. It is worth noticing that here, with respect to analogous critical point theorem, such local minimum result is obtained by using a new type of Palais-Smale condition introduced in [\textit{G. Bonanno}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75, No. 5, 2992--3007 (2012; Zbl 1239.58011)].NEWLINENEWLINEFor a complete overview on these topics see: [\textit{B. Ricceri}, Arch. Math. 75, No. 3, 220--226 (2000; Zbl 0979.35040); J. Comput. Appl. Math. 113, No. 1--2, 401--410 (2000; Zbl 0946.49001)], [\textit{S. A. Marano} and \textit{D. Motreanu}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 48, No. 1, 37--52 (2002; Zbl 1014.49004); J. Differ. Equations 182, No. 1, 108--120 (2002; Zbl 1013.49001)], [\textit{G. Bonanno} and \textit{P. Candito}, J. Differ. Equations 244, No. 12, 3031--3059 (2008; Zbl 1149.49007)].