Semilinear partial differential equations of constant strength (Q2704050)

Semilinear operators of the type NEWLINE\[NEWLINE F(u)=P(D)u+f(x,Q_1(D)u,\dots,Q_M(D)u) NEWLINE\]NEWLINE are studied where \(P(D)\), \(Q_1(D),\dots,Q_M(D)\) are linear partial differential operators in \(R^n\) with constant coefficients and \(f(x,v)\) \((x\in R^n\), \(v\in C^M)\) is a smooth function with respect to \(x\) and an entire function with respect to \(v\). It is assumed that \(f(x_0,v)=0\) for a fixed \(x_0\) and all \(v\in C^M\). The existence of a solution to the equation \(F(u)=g\) in a neighbourhood of \(x_0\) for an arbitry \(g\in B_{2,k}\) is proved. Here \(B_{2,k}\) is the weight Sobolev space introduced by L. Hörmander with the temperate weight function \(k\) satisfying certain assumptions. The proof is based on the algebra property of the spaces \(B_{2,k}\) and on the fixed point theorem.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00030].