Existence of solution for two classes of elliptic problems in \(\mathbb R^N\) with zero mass (Q414834)

The existence of a positive solution to the following elliptic equation is obtained: NEWLINE\[NEWLINE -\Delta u = K(x)f(u) \quad \text{in } \mathbb{R}^n; NEWLINE\]NEWLINE where \(f\) is continuous and has quasi-critical growth. Two different classes of function \(K\) are considered: The first case concerns positive continuous maps \(K\) which are asymptotically periodic as \(|x| \to \infty\). The second case concerns maps \(K \in L^\infty\cap L^r\) positive almost everywhere.NEWLINENEWLINEThe proofs rely on variational methods. The associated energy functional satisfies the geometry of the mountain pass.