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Regularity and existence of solutions of elliptic equations with p,q- growth conditions. (English) Zbl 0724.35043

This paper deals with weak solutions of ellipitic equations of the form: \[ (1.1)\quad \sum^{n}_{i=1}\partial_{x_ i}a^ i(x,Du)=b(x),\quad x\in \Omega, \] where \(\Omega\) is an open subset of \({\mathbb{R}}^ n\) and \(a^ i\) satisfy some nonstandard growth conditions: \[ \sum a^ i_{S_ i}(x,\xi)\lambda_ i\lambda_ j\geq m(1+| \xi |^ 2)^{(p-2)/2}| \lambda |^ 2,\quad | a^ i_{\xi_ i}(x,\xi)| \leq M(1+| \xi |^ 2)^{(q- 2)/2},\quad q\geq p\geq 2. \] The first is a regularity result: every weak solution to (1.1) of class \(W^{1,q}_{loc}(\Omega)\) is locally Lipschitz continuous in \(\Omega\). A second type of result concerns the existence of solutions to equation (1.1) satisfying some given Dirichlet boundary conditions: the “a priori” regularity results previously stated are applied here.
Reviewer: M.A.Vivaldi (Roma)

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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