A note on solutions of two-dimensional semilinear elliptic vector-field equations with strong nonlinearity

We prove the existence of solutions u for some semilinear elliptic vector-field equations on IR2 with a nonlinearity which is allowed to grow at infinity ‘nearly like a linear exponential’ in lul. In some cases a growth like exp(alulγ), a > 0 , 0 < γ < 2 , is allowed. This is achieved by introducing an appropriate space E of ‘à priori-solutions’ for which some important continuous imbeddings are proven replacing the well known Sobolev-imbeddings for the d ≥ 3-dimensional case. Then the standard variational method is applied.