Semilinear elliptic Neumann problems with rapid growth in the nonlinearity

Abstract

The existence and regularity of solutions to semilinear elliptic Neumann problems are investigated. Motivated by the Poisson–Boltzmann equation of biophysics and semiconductor modeling, the nonlinearity is assumed to be a continuous, strictly monotone increasing function that passes through the origin with asymptotically superlinear and unbounded growth. Pseudomonotone operator theory is utilised to establish the existence and uniqueness of a weak solution in the Sobolev space W1,2. With an additional assumption on the nonlinearity, we show that this weak solution belongs to .