Positive solutions for elliptic problems with critical indefinite nonlinearity in bounded domains

Abstract

In this paper, we study the semilinear elliptic problem with critical nonlinearity and an indefinite weight function, namely $$- Delta u =lambda u + h (x) u^{(n+2)/(n-2)}$$ in a smooth open bounded domain $Omegasubseteq mathbb{R}^n$, $n > 4$ with Dirichlet boundary conditions and for $lambda geq 0$. Under suitable assumptions on the weight function, we obtain the positive solution branch, bifurcating from the first eigenvalue $lambda_1(Omega)$. For $n=2$, we get similar results for $-Delta u =lambda u + h (x)phi(u)e^u$ where $phi$ is bounded and superlinear near zero