Quasilinear ellipticity and the Dirichlet problem
For Dirichlet problems of the form \[ L(u)=-\sum D_ i(a_{ij}(x,u)D_ j(u))=f(u)+h(x) \] existence and nonexistence results in suitable spaces are proved, depending on the interaction of the nonlinearity f with the spectrum of elliptic operators, which are associated with the above mentioned L. As an example, let the coefficients of L have uniform limits as \(| u| \to \infty\), then with these limit coefficients such an elliptic operator is given [look also at earlier papers of the author, for example J. Anal. Math. 50, 65-78 (1988) or Rend. Circ. Mat. Palermo, II. Ser. 35, 159-168 (1986; Zbl 0645.35028)]. In the proofs, homotopy- arguments, degree theory, \(W^ 1_ 2\)-theory, sub- and super-solutions and G-convergence are used. It is remarkable, that in some cases, the boundary values can be in \(L_ 2(\partial Q)\). All results are embedded in the corresponding literature, f.e., \textit{V. L. Shapiro} [J. Differ. Equations 68, 253-280 (1987; Zbl 0629.35043)].
scientific article; zbMATH DE number 4161227
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