Asymptotic boundary estimates for solutions to the \(p\)-Laplacian with infinite boundary values. (English) Zbl 1524.35268
Summary: In this paper, by using Karamata regular variation theory and the method of upper and lower solutions, we mainly study the second order expansion of solutions to the following \(p\)-Laplacian problems: \(\Delta_p u=b(x)f(u), u>0, x\in \Omega, u|_{\partial \Omega}=\infty\), where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^N\) \((N\geq 2), p>1, b \in C^{\alpha}(\bar{\Omega})\) which is positive in \(\Omega\) and may be vanishing on the boundary. The absorption term \(f\) is normalized regularly varying at infinity with index \(\sigma >p-1\). The results extend some previous findings of D. Repovš [J. Math. Anal. Appl. 395, No. 1, 78–85 (2012; Zbl 1250.35109)] in a certain sense.
MSC:
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
\(p\)-Laplacian problems; second expansion of solutions; upper and lower solutions; Karamata regular variation theoryCitations:
Zbl 1250.35109References:
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