On indefinite superlinear elliptic equations
The author studies the solvability of the semilinear elliptic Neumann problem \[ - \Delta u+ m(x) u= a(x) u^p,\quad u> 0\quad \text{in } \Omega,\quad \partial u/\partial n= 0\quad \text{on }\partial\Omega,\tag{1} \] \(\Omega\subset \mathbb{R}^n\) a bounded, smooth domain. He assumes that \(p\) is subcritical, \(m\) is Hölder continuous and \(a\) is continuous and may change sign. Denote by \(\lambda_1\) the first eigenvalue of the linear operator \(-\Delta+ m(x)\) with homogeneous Neumann boundary condition, and by \(\phi\) the corresponding positive eigenfunction normalized to \(\int \phi^2= 1\). A typical result of this paper is the following: Assume, \(m\) is bounded in the Hölder norm, \(\int a(x) dx< 0\), \(\int m(x) dx\leq 0\); then there exists a constant \(\Lambda\) (only depending on the norm of \(m\)), such that if \(0> \lambda_1> \Lambda\) and \(\int a\phi^{p+ 1} dx< 0\), problem (1) has a solution. It is shown by a counterexample, that the condition \(\int a(x) dx< 0\) is essential for this existence result. The same problem was studied recently by \textit{H. Berestycki}, \textit{I. Capuzzo-Dolcetta} and \textit{L. Nirenberg} [NoDEA 2, No. 4, 553-572 (1995; Zbl 0840.35035)] in the case \(\lambda_1\geq 0\).
scientific article; zbMATH DE number 852857