The convergence of nonnegative solutions for the family of problems \(-\varDelta_p u= \lambda e^u\) as \(p \rightarrow \infty\). (English) Zbl 1404.35311
The paper is concerned with the problem
\[
\begin{cases} -\Delta_pu=\lambda e^u, &\text{for }x\in\Omega,\\ u=0, &\text{for }x\in\partial\Omega,\end{cases}\tag{1}
\]
where \(\Omega\subset \mathbb{R}^N\) (\(N\geq 2\)) is a bounded domain with smooth boundary \(\partial \Omega\), \(p>N\), and \(\Delta_p=\text{div}(|\nabla|^{p-2}\nabla)\) is the \(p\)-Laplace operator. The authors prove the existence of a positive real number \(\lambda^*\) such that for each \(\lambda\in (0,\lambda^*)\) problem (1) has a nonnegative weak solution \(u_p\). The uniform convergence of \(u_p\) as \(p\to\infty\) to the distance function to the boundary of the domain is also shown.
Reviewer: Rodica Luca (Iaşi)
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 35J62 | Quasilinear elliptic equations |
| 35D30 | Weak solutions to PDEs |
| 35D40 | Viscosity solutions to PDEs |
| 47J30 | Variational methods involving nonlinear operators |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
weak solution; viscosity solution; nonlinear elliptic equations; asymptotic behavior; distance function to the boundaryReferences:
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