Nonlinear distributions and quasilinear ellipticity (Q1093813)

Consider in a bounded domain \(\Omega \subset R^ N\) the equation \[ Q(u)\equiv -D_ i[a^{ij}(x,u)D_ ju+b^ i(x,u)u]+c^ i(x,u)D_ iu+d(x,u)u=f(x,u)+g_ 0(x,u)-D_ ig_ i(x,u) \] where functions \(a^{ij}\), \(b^ i\), \(c^ i\), d, \(g_ i\) satisfy a linear growth condition in u and uf(x,u) may be arbitrary quickly converging to - \(\infty\) as \(u\to \pm \infty\). It is proved the existence of a distribution solution \(u\in H_ 0^ 1(\Omega)\) of this equation with f(x,u), \(f(x,u)u\in L^ 1(\Omega)\). Further, by some additional conditions it is shown the existence of a distribution solution \(u\in H^ 1_ 0(\Omega)\) of the equation at resonance \[ Q(u)=\lambda^*_ 1u+f(x,u)+g_ 0(x,u)-D_ ig_ i(x,u) \] with f(x,u), \(uf(x,u)\in L^ 1(\Omega)\) where \(\lambda^*_ 1\) is the generalized first eigenvalue of Q.