On an Emden-Fowler type equation. (English) Zbl 0781.35021
The authors study the problem:
\[
\Delta_ p u=- \text{div}(|\text{grad } u|^{p-2}\text{grad }u)= \lambda e^ u\text{ in }\Omega,\;u=0\text{ on }\partial\Omega,
\]
where \(\lambda>0\) and \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^ n\). If \(1<p<n\), then there is a constant \(\lambda'\) such that for \(0<\lambda<\lambda'\) the problem has a solution. If \(p\geq n\), then for small \(\lambda\) there exist at least two solutions. There is \(\lambda_ 0\) such that for \(\lambda<\lambda_ 0\) the problem has no solutions. If \(\Omega=\mathbb{R}^ n\), then for all \(\lambda>0\) the solution does not exist in the class \(W_{1,p}(\mathbb{R}^ n)\).
Reviewer: Yu.V.Egorov (Toulouse)
MSC:
| 35J70 | Degenerate elliptic equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35D10 | Regularity of generalized solutions of PDE (MSC2000) |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
Keywords:
nonlinear degenerate elliptic problems; comparison principle; supersolution; weak solutions; Sobolev space; existence of regular solutionsReferences:
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