A priori estimates and existence for quasi-linear elliptic equations. (English) Zbl 1169.35336

Zou, Heng Hui

Calc. Var. Partial Differ. Equ. 33, No. 4, 417-437 (2008).

Summary: We study the boundary value problem for quasi-linear elliptic equation
\[ \begin{aligned}{\text{div}}(|\nabla u|^{m-2} \nabla u) + B(z,u,\nabla u) &= 0 \quad \text{in }\Omega,\\ u &= 0 \quad \text{on } \partial\Omega, \end{aligned} \] where \(\Omega\subset\mathbb R^n\) \((n\geq 2)\) is a connected smooth domain, and the exponent \({m\in(1,n)}\) is a positive number. Under appropriate conditions on the function \(B\), a variety of results on a priori estimates, existence and non-existence of positive solutions are established. The results are generically optimum for the canonical prototype \(B = |u|^{p-1} u\), \(p>m-1\).

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35J55	Systems of elliptic equations, boundary value problems (MSC2000)
35B45	A priori estimates in context of PDEs
35J20	Variational methods for second-order elliptic equations

Keywords:

quasi-linear elliptic equation; a priori estimates; Liouville theorem; existence

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