Elliptic equations with decaying cylindrical potentials and power-type nonlinearities
The authors study the following cylindrical nonlinear elliptic equation: \[ -\Delta u+\frac{A}{| y| ^\alpha}u=f(u), \] where \(x=(y,z)\in \mathbb{R}^k\times\mathbb{R}^{N-k}\) (\(2\leq k<N\)), \(A, \alpha >0\), and \(f\) is a nonlinearity of power growth \(p\). Main results concern existence, nonexistence and asymptotic behavior of solutions, and rely on certain compatibility conditions between the exponents \(\alpha\) and \(p\). The proof of existence uses critical point theory.
scientific article; zbMATH DE number 5343713