Existence and uniqueness of solutions for some degenerate nonlinear elliptic equations. (English) Zbl 1324.35039
The author proves the existence and uniqueness of a weak solution in certain weighted Sobolev spaces for the Dirichlet problem
\[
\begin{aligned} L u(X) &=f_0(x)-\sum_{j=1}^n D_j f_j(x) \text{ in }\Omega, \\ u(x) &=0 \text{ on }\partial \Omega, \end{aligned}
\]
with the partial differential operator \(L\) defined as
\[
L u(x)=\Delta \Bigl( v(x) |\Delta u|^{p-2} \Delta u\Bigr) - \sum_{j=1}^n D_j [\omega (x) \mathcal{A}_j(x, u(x), \nabla u(x))].
\]
The set \(\Omega\) is a bounded open set, \(\omega\) and \(v\) are two weight functions of the Muckenhoupt class, \(1<p<\infty\), and the functions \(\mathcal{A}_j\) satisfy certain regularity and growth assumptions. The author proves by a topological argument (uniqueness of solution of the equation \(Ax=T\) with strictly monotone, coercive, and hemicontinuous operator \(A\) on the real, separable, reflexive Banach space) that the problem has a unique solution and provides an estimate for its norm.
Reviewer: Robert Mařík (Brno)
MSC:
35J70 | Degenerate elliptic equations |
35J60 | Nonlinear elliptic equations |
35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |