Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

Abstract.

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

$$ (P)\left\{ {\begin{array}{*{20}l} {{ - \Delta u = \lambda u + h(x)\phi (u)e^{u} {\text{ in }}\mathbb{R}^{2} }} \\ {{u \geq 0\,\,u \to 0\,{\text{ when }}||x||\, \to + \infty }} \\ \end{array} } \right. $$

We consider a function h which is smooth and changes sign.