Boundary value problems for local and nonlocal Liouville type equations with several exponential type nonlinearities. Radial and nonradial solutions

Abstract

AbstractThis paper deals with boundary value problems for local and nonlocal Laplace operator in 2D with exponential nonlinearities, the so-called Liouville type equations. They include the mean field equation and other equations arising in the statistical mechanics. Existence results into an explicit form for the Dirichlet problem in the unit disc $B_{1} \subset {\mathbf{R}}^{2} $ B 1 ⊂ R 2 and in the participation of positive parameters in the right-hand sides are proved in Theorems 2 and 3. Theorem 2 is illustrated by several examples including an application to the differential geometry. In Theorem 4 global radial solution of the Cauchy problem with constant data at $\partial B_{1} $ ∂ B 1 and under appropriate conditions is constructed. It develops logarithmic singularities for $r = 0 $ r = 0 , $r = \infty $ r = ∞ . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper.