Abstract:
In this paper, we study the first boundary value problem for $p(x)$-Laplacian with one spatial variable in the presence of gradient terms that do not satisfy the Bernstein—Nagumo condition. A class of gradient nonlinearities is defined, for which the existence of a viscosity solution that is Lipschitz continuous in $x$ and Hölder continuous in $t$ is proven.
Keywords:$p(x)$-Laplace equation, Bernstein—Nagumo type condition, viscosity solutions, a priori estimates.
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