I'm dealing with a nonnegative functional: $$E(\phi)=\int_{B_R}\left[|\nabla \phi|^2-f^{\prime}(u) \phi^2\right] \geq 0, \text{for any} \ \phi \in C_0^{\infty}\left(B_R\right).$$
The paper told me that since $E(\phi) \geq 0$ for all $\phi \in C_0^{\infty}\left(B_R\right)$, so by variational characterization of eigenvalues, there are a $\lambda_R \geq 0$ and a positive eigenfunction $\Phi_R$ on $B_R$ satisfying Dirichlet boundary conditions on $\partial B_R$ so that $$ -\Delta \Phi_R-f^{\prime}(u) \Phi_R=\lambda_R \Phi_R . $$ I want to know what the reference this step needs. If we integrate $E(\phi)$ by part we get: $$E(\phi)=\int_{B_R}\left[(-\Delta \phi-f^{\prime}(u) \phi)\phi\right] \geq 0, \text{for any} \ \phi \in C_0^{\infty}\left(B_R\right).$$ Then how do we get the eigenvalue and eigenfunction which are both positive?