Abstract
This 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} 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} and under appropriate conditions is constructed. It develops logarithmic singularities for r = 0 , r = infty . An illustrative example to Theorem 4 in the case of two exponents is given at the end of the paper.
Highlights
Following the pioneering work [15] of L
In what follows we propose several papers on the subject [2,3,4, 6, 9, 16, 19, 25,26,27, 30]
Very often in investigating these boundary value problems (BVP) variational methods are applied. The advantage of this approach is that one can work in bounded smooth domains in the plane
Summary
Following the pioneering work [15] of L. Very often in investigating these boundary value problems (BVP) variational methods are applied (for example, see [12, 14]). The advantage of this approach is that one can work in bounded smooth domains in the plane. Our aim here is to find (mainly) radial solutions to the boundary value problems for local and nonlocal PDE of Liouville type. Slavova and Popivanov Advances in Difference Equations (2021) 2021:386 methods are applied in the investigation of elliptic PDE with exponential nonlinearities, see [13]. We shall consider two nonlocal BVPs which do not possess radial solutions
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