Solvability of Dirichlet problems for semilinear elliptic equations on certain domains
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), \(n\geq3\), \(|\Omega |\) be the volume of the domain. Consider the Dirichlet problem (P) \(-\Delta u= f(x, u)\) in \(\Omega\); \(u=0\) on \(\partial \Omega\), where \(\Delta\) is the standard Laplace operator, \(f(x, u)\) is a local Hölder continuous function defined on \(\overline {\Omega} \times \mathbb{R}\). The main result of the paper is as follows: Assume that there are positive constants \(M_1\), \(M_2\), \(q\geq 1\), such that \(|f(x, t)|\leq M_1+ M_2|t|^q\) for all \(x\in \overline {\Omega}\), \(t\in \mathbb{R}\); then there is a constant \(c(n, q)\) depending only on \(n\), \(q\), such that if \(|\Omega |\leq c(n, q) (M_2 M_1^{q-1} )^{-{n \over 2q}}\), the problem (P) has at least one solution. There is no upper bound restriction on \(q\) in the result. Thus this result improves an earlier similar result obtained by the author [Indiana Univ. Math. J. 42, 259-282 (1993; Zbl 0797.35049)]. The method used here is a combination of the upper and lower solution method and the variational method. A change of variables is used to construct a pair of upper and lower solutions.
scientific article; zbMATH DE number 950199