A priori bounds for an indefinite superlinear elliptic equation with exponential growth (Q2465885)

This paper deals with the positive solutions of the following semilinear elliptic partial differential equation: \[ -\Delta u=\lambda u+ h(x) e^u,\quad x\in\mathbb{R}^2,\tag{1} \] where \(\lambda\in\mathbb{R}\) is a parameter and the function \(h\in C^2\) changes its sign along a \(C^2\) manifold of dimension 1. Under several restrictions imposed on \(h\) the author proves that for any positive \(K\) such that \(\lambda>-K\) the positive solutions of the equation (1) are uniformly bounded from above. In proving this result the author has adapted the techniques used in \textit{W. Chen} and \textit{C. Li} [J. Differ. Equ. 195, No. 1, 1--13 (2003; Zbl 1134.35331)].