Limits as \(p\rightarrow\infty \) of \(p\)-Laplacian problems with a superdiffusive power-type nonlinearity: positive and sign-changing solutions. (English) Zbl 1198.35115
Summary: We investigate the asymptotic behaviour as \(p\to\infty\) of sequences of solutions of the equation
\[ -\Delta_pu= \lambda|u|^{q(p)-2}u \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega, \]
where \(\lambda>0\) and \(q(p)>p\) with \(\lim_{p\to\infty} q(p)/p= Q\geq 1\). We are interested in the characterization of such limits as viscosity solutions of a PDE problem. Both positive and sign-changing solutions are considered.
\[ -\Delta_pu= \lambda|u|^{q(p)-2}u \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega, \]
where \(\lambda>0\) and \(q(p)>p\) with \(\lim_{p\to\infty} q(p)/p= Q\geq 1\). We are interested in the characterization of such limits as viscosity solutions of a PDE problem. Both positive and sign-changing solutions are considered.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J62 | Quasilinear elliptic equations |
| 35D40 | Viscosity solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B09 | Positive solutions to PDEs |
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