Positive solution branch for elliptic problems with critical indefinite nonlinearity

Abstract

In this paper, we study the semilinear elliptic problem with critical nonlinearity and an indefinite weight function, namely $- \Delta u =\lambda u + h (x) u^{\frac{n+2}{n-2}} $ in a smooth domain bounded (respectively, unbounded) $\Omega\subseteq\,\mathbb R^n , \ n > 4 $, for $\lambda \geq 0 $. Under suitable assumptions on the weight function, we obtain the positive solution branch, bifurcating from the first eigenvalue $\lambda_1(\Omega)$ (respectively, the bottom of the essential spectrum).