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We study multidimensional backward stochastic differential equations (BSDEs)which cover the logarithmic nonlinearity u log u. More precisely, we establishthe existence and uniqueness as well as the stability of p-integrable solutions(p > 1) to multidimensional BSDEs with a p-integrable terminal condition and asuper-linear growth generator in the both variables y and z. This is done witha generator f(y, z) which can be neither locally monotone in the variable y norlocally Lipschitz in the variable z. Moreover, it is not uniformly continuous.As application, we establish the existence and uniqueness of Sobolev solutionsto possibly degenerate systems of semilinear parabolic PDEs with super-lineargrowth generator and an p-integrable terminal data. Our result cover, forinstance, certain (systems of) PDEs arising in physics.

p-integrable solutions to multidimensional BSDEs and degenerate systems of PDEs with logarithmic nonlinearities