On the Neumann \((p, q)\)-eigenvalue problem in Hölder singular domains. (English) Zbl 1543.35121
Summary: In the article we study the Neumann \((p, q)\)-eigenvalue problems in bounded Hölder \(\gamma\)-singular domains \(\Omega_\gamma\subset\mathbb{R}^n\). In the case \(1 < p < \infty\) and \(1 < q < p^\ast_\gamma\) we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of \((p, q)\)-eigenvalues.
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |
Keywords:
regularity results of eigenfunctionsReferences:
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