An existence theorem for quasilinear elliptic equations on the \(N\)-torus (Q1201205)

The author proves the existence of a distributional solution \(u \in W^{m,2} (\Omega)\) of the equation \[ Qu \equiv (- 1)^{|\beta |} D^\beta \bigl( a_{\alpha \beta} (x, Du) D^\alpha u \bigr) = g(x,u) - h, \;\bigl( 1 \leq |\alpha |, |\beta |\leq m \bigr) \] (where \(h \in W^{- m,2} (\Omega))\), under the condition \(\int_\Omega g_+ (x)dx < h(1) < \int_\Omega g_- (x) dx\), where \(g_\pm\) are defined by \(g_+ (x) = \lim \sup_{s \to \infty} g(x,s)\), \(g_- (x) = \lim \inf_{s \to \infty} g(x,s)\). Here, \(\Omega\) is the \(N\)-dimensional Torus.