Weak solutions for a class of quasilinear elliptic equation containing the $p(· )$-Laplacian and the mean curvature operator in a variable exponent Sobolev space - Junichi Aramaki (Electronic Journal of Qualitative Theory of Differential Equations)
In this paper, we consider the equation for a class of nonlinear operators containing $p(· ) $-Laplacian and mean curvature operator with mixed boundary conditions in a bounded domain $Ω $ of $\mathbb{R}^N$, under the hypothesis $p(x)>1$ in $\overline{Ω}$. More precisely, we are concerned with the problem under the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of one, two and infinitely many nontrivial weak solutions of the equation according to the conditions on given functions.