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We consider the mathematical treatment of a system of nonlinear partial differential equations based on a model, proposed in 1972 by J. Newman, in which the coupling between the Lithium concentration, the phase potentials and temperature in the electrodes and the electrolyte of a Lithium battery cell is considered. After introducing some functional spaces well-adapted to our framework we obtain some rigorous results showing the well-posedness of the system, first for some short time and then, by considering some hypothesis on the nonlinearities, globally in time. As far as we know, this is the first result in the literature proving existence in time of the full Newman model, which follows previous results by the third author in 2016 regarding a simplified case. Keywords: Lithium-ion battery cell, multiscale mathematical model, Green operators, fixed point theory, Browder-Minty existence results, super and sub solutions 2010 MSC: 35M10, 35Q60, 35C15, 35B50, 35B60• The electric potential φ s = φ s (x, t) in the electrodes.• The electric potential measured by a reference Lithium electrode in the electrolyte, ϕ e = ϕ e (x, t).• The temperature T (t) in the cell.The system of equations is given by (5)-(9) below.

We prove the exact multiplicity of flat and compact support stable solutions of an autonomous non-Lipschitz semilinear elliptic equation of eigenvalue type according to the dimension N and the two exponents, 0 < α < β < 1, of the involved nonlinearites. Suitable assumptions are made on the spatial domain Ω where the problem is formulated in order to avoid a possible continuum of those solutions and, on the contrary, to ensure the exact number of solutions according to the nature of the domain Ω. Our results also clarify some previous works in the literature. The main techniques of proof are a Pohozhaev's type identity and some fibering type arguments in the variational approach.for any non-zero weak solution w λ of P (α, β, λ). Notice that in [23] authors also use the term "ground state" with a different meaning.Since the diffusion-reaction balance −∆u = f (λ, u) involves the non-linear reaction termand it is a non-Lipschitz function at zero (since α < 1 and β < 1) important peculiar behavior of solutions of these problems arises. For instance, that may lead to the violation of the Hopf maximum principle on the *

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