Nonlinear elliptic equations having a gradient term with natural growth. (English) Zbl 1158.35364
The authors deal with a class of nonlinear elliptic Dirichlet problems whose simplest model example is
\[
-\Delta_p u= g(x)|\nabla u|^p+ f\quad\text{in }\Omega,
\]
\[ u= 0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded open set in \(\mathbb{R}^N\), \(N\geq 2\), \(g\) is a continuous real function and \(\Delta_p\) denotes the so-called \(p\)-Laplacian \((p> 1)\). Under some natural assumptions on \(g\), \(f\) the authors prove the \(L^\infty\)-estimate and deduce the existence of bounded solutions. Moreover, the authors study the existence of unbounded solutions (and their regularity); in this context they use the more general framework of entropy solutions, which can be applied to the full range \(f\in L^m(\Omega)\), \(m> 1\). Furthermore, the authors present several examples concerning the optimality their results in the range of assumptions appeared in the paper.
\[ u= 0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded open set in \(\mathbb{R}^N\), \(N\geq 2\), \(g\) is a continuous real function and \(\Delta_p\) denotes the so-called \(p\)-Laplacian \((p> 1)\). Under some natural assumptions on \(g\), \(f\) the authors prove the \(L^\infty\)-estimate and deduce the existence of bounded solutions. Moreover, the authors study the existence of unbounded solutions (and their regularity); in this context they use the more general framework of entropy solutions, which can be applied to the full range \(f\in L^m(\Omega)\), \(m> 1\). Furthermore, the authors present several examples concerning the optimality their results in the range of assumptions appeared in the paper.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
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