Solvability of boundary-value problems for quasilinear elliptic and parabolic equations in unbounded domains in classes of functions growing at infinity - Shishkov, A. E. (Ukrains’kyi Matematychnyi Zhurnal, 1995)
For divergent elliptic equations with the natural energetic spaceW p m (Ω),m≥1,p>2, we prove that the Dirichlet problem is solvable in a broad class of domains with noncompact boundaries if the growth of the right-hand side of the equation is determined by the corresponding theorem of Phragmén-Lindelöf type. For the corresponding parabolic equation, we prove that the Cauchy problem is solvable for the limiting growth of the initial function % MathType!MTEF!2!1!+- u_0 (x) L_2.loc (R^n ): _|x|< τ u_0^2 dx cτ ^n + 2mp/(p - 2) τ< ∞