Abstract:
We consider the third boundary value problem for a p-Laplace equation with a low-order term that does not satisfy the Bernstein–Nagumo condition. The solvability of the problem in the class of radially symmetric solutions is investigated. A class of gradient nonlinearities is defined, for which the existence of a weak Sobolev radially symmetric solution with a Holder continuous derivative with exponent 1/p-1 is proven. It is shown that nonlinearity in the gradient can be arbitrary, provided that the low order term containing the gradient is Lipschitz continuous in the spatial variable and strictly monotone in the variable u. The solution to the original problem is approximated by classical solutions to the corresponding regularized problem. The a priori estimates obtained for the regularized problem do not depend on the regularization parameter, which allows us to obtain a solution to the original problem of the specified smoothness by passing to the limit.
Keywords:p-Laplace equation, Bernstein–Nagumo condition, radially symmetric solutions, a priori estimates
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