Remarks on Dirichlet problems with sublinear growth at infinity (Q2809915)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with sufficiently regular boundary \(\partial \Omega\). The authors consider the following parameterized Dirichlet problem NEWLINE\[NEWLINE (P_\lambda)\quad -\Delta u= \lambda a(x)g(u)\text{ in }\Omega,\quad u=0\text{ on }\partial \Omega,NEWLINE\]NEWLINE where \(\lambda \in (0,+\infty)\), \(g:\mathbb{R}_+\rightarrow \mathbb{R}_+\) is a continuous function such that \(g(0)=0\) and \(g(t)>0\) if \(t>0\), and \(a\in L^\infty(\Omega)\setminus \{0\}\) is a weight which is allowed to be sign-changing. The function \(g\) is assumed to have linear growth at \(0\) and sublinear growth at \(\infty\).NEWLINENEWLINEThe authors present some existence and uniqueness results for the above problem which are consequences of well known Brezis-Oswald and Brown-Hess Theorems (see [\textit{H. Brézis} and \textit{L. Oswald}, Nonlinear Anal., Theory Methods Appl. 10, 55--64 (1986; Zbl 0593.35045)] and [\textit{K. J. Brown} and \textit{P. Hess}, Differ. Integral Equ. 3, No. 2, 201--207 (1990; Zbl 0729.35046)]). The optimality of some hypotheses is also investigated by exhibiting appropriate examples. Finally, the authors study the structure of the set of the positive solution pairs (i.e. the set of pairs \((\lambda,u)\), with \(u\) positive solution to \((P_\lambda)\)) by using the time-mapping technique in the one dimensional case and Rabinowitz's global bifurcation theorem in higher dimension.