## On multiquasielliptic equations in $$\mathbb R_n$$.(Russian, English)Zbl 1035.35028

Sib. Mat. Zh. 44, No. 5, 1183-1188 (2003); translation in Sib. Math. J. 44, No. 5, 926-930 (2003).
The author investigates the following partial differential equation in $$\mathbb R_n$$: $P(D_x)u = f(x),$ where $P(D_x) = \sum_{\gamma\in M_P}a_{\gamma}D_x^{\gamma}$ is the multiquasielliptic operator with constant coefficients, $$M_P$$ denotes the finite set of multi-indexes $$\gamma = (\gamma_1,\dots,\gamma_n)$$, $$D_x^{\gamma} = D_1^{\gamma_1}\cdots D_n^{\gamma_n}$$, $$D_j = -i\partial/\partial x_j$$. Under some reasonable assumptions on the characteristic polynomial associated with the differential operator $$P(D_x)$$, the author proves the existence of a generalized solution to the equation under consideration in Sobolev-type spaces. To prove the existence result, the author use the technique of integral operators and a priori estimates.

### MSC:

 35H30 Quasielliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35B45 A priori estimates in context of PDEs 35D10 Regularity of generalized solutions of PDE (MSC2000)

### Keywords:

existence theorem; generalized solution; Sobolev-type space
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