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A priori bounds for weak solutions to elliptic equations with nonstandard growth. (English) Zbl 1261.35061

In this paper, by using De Giorgi’s iteration method and the localization method, the authors establish a priori bounds for sub- and supersolutions, in particular for solutions of some elliptic equations with variable exponents of the form \[ \begin{aligned} -\text{div}\,&\mathcal A(x, u, \nabla u)=\mathcal B(x, u, \nabla u)\quad\text{in}\,\, \Omega\\ &\mathcal A(x, u, \nabla u).\,\nu=\mathcal C(x, u)\quad \text{on}\,\,\partial \Omega, \end{aligned} \] where \(\nu(x)\) is the outer unit normal of \(\Omega\) at \(x\in \partial \Omega,\) and \(\mathcal A, \mathcal B:\Omega \times \mathbb R \times \mathbb R^N\to \mathbb R\) and \(\mathcal C: \partial\Omega \times \mathbb R \to \mathbb R\) are Carathéodory functions satisfying the following \(p(x)\)-structure conditions: \[ |\mathcal A(x, s, \xi)|\leq a_0|\xi|^{p(x)-1}+a_1| s |^{q_0(x)\frac{p(x)-1}{p(x)}}+a_2, \qquad \text{for a.a.}\quad x\in\Omega,\tag{1} \]
\[ \mathcal A(x, s, \xi).\xi\geq a_3|\xi|^{p(x)}-a_4| s|^{q_0(x)}-a_5, \qquad\text{for a.a.}\quad x\in\Omega,\tag{2} \]
\[ |\mathcal B(x, s, \xi)|\leq b_0|\xi|^{p(x)\frac{q_0(x)-1}{q_0(x)}}+b_1| s|^{q_0(x)-1}+b_2, \quad\,\, \text{for a.a.}\quad x\in\Omega,\tag{3} \]
\[ |\mathcal C(x, s)|\leq c_0| s|^{q_1(x)-1}+c_1, \qquad \text{for a.a.}\quad x\in\partial\Omega,\tag{4} \] for all \(s\in \mathbb R\) and all \(\xi \in \mathbb R^N.\) Here \(a_i, b_j\) and \(c_l\) are positive constants, \(p\in C(\overline{\Omega})\) with \(\inf_{\overline{\Omega}}p(x)>1, q_0\in C(\overline{\Omega})\) and \(q_1\in C({\partial\Omega})\) are chosen such that \[ p(x)\leq q_0(x)<p^\ast(x), \quad x\in \overline{\Omega},\,\, \text{and}\,\,\,p(x)\leq q_1(x)<p_\ast(x), \quad x\in \partial\Omega. \] \(p^\ast\) and \(p_\ast\) are the usual critical exponents.

MSC:

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
35D30 Weak solutions to PDEs
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