Minimal potential results for Schrödinger equations with Neumann boundary conditions. (English) Zbl 1379.35146
Summary: We consider the boundary value problem \(-\Delta_{p}u=V|u|^{p-2}u-C\), where \(u\in W^{1,p}(D)\) is assumed to satisfy Neumann boundary conditions, and \(D\) is a bounded domain in \(\mathbb{R}^{n}\). We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for the product of a sharp Sobolev constant and an \(L^{p}\) norm of \(V\). When \(p=n\), Orlicz norms are used. In many cases, these inequalities are best possible. Applications to linear and non-linear eigenvalue problems are also discussed.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35P15 | Estimates of eigenvalues in context of PDEs |
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