Some observations on the regularity of the solutions of strongly nonlinear problems and applications
The authors study at first the regularity of solutions of the strongly nonlinear elliptic problem \[ u\in W_ 0^{1,p}(\Omega),\quad g(.,u)\in L^ 1(\Omega),\quad ug(.,u)\in L^ 1(\Omega), \] \[ <Au- f,v>+\int_{\Omega}g(.,u)v=0;\quad \forall v\in W_ 0^{1,p}(\Omega)\cap L^{\infty}(\Omega)\quad and\quad for\quad v=u; \] where \(\Omega\) is a bounded domain in \(R^ N\), A is a strongly nonlinear elliptic operator of second order and growth p \((p>1)\) and g is a Carathéodory function satisfying one-sided condition \(g(x,\eta)\eta \geq 0\) and sup\(\{g(x,\eta)\); \(| \eta | <\lambda\} = h_{\lambda}(x)\in L^ 1(\Omega).\) Let \(f=-\sum (f_ i)_{x_ i}\), \(f_ i\in L^ q(\Omega)\), \(p\leq N\). It is proved that for \(q>N/(p-1)\) the solution u is bounded on \(\Omega\) and for \(p'\leq q<N/(p-1)\) the solution u belongs to \(L^ s(\Omega)\) where s is the exponent of Sobolev embedding theorem corresponding to q(p-1). Under additional conditions on g, the authors prove higher integrability of g(u). Moreover, they extend the previous results to variational inequalities with an obstacle \(\psi\in W_ 0^{1,p}(\Omega) \cap L^{\infty}(\Omega)\). The regularity results are used to prove the following assertion on homogenization: Let strongly nonlinear operators \(A_{\epsilon}\) H-converge to A, \(c_{\epsilon}\) converge to \(c_ 0\) in \(\sigma(L^{\infty}(\Omega),L^ 1(\Omega))\). Then (under some assumptions on \(A_{\epsilon},c_{\epsilon})\) solutions \(u_{\epsilon}\) of the problem \[ A_{\epsilon}u_{\epsilon} + c_{\epsilon}(x) g(x,u_{\epsilon}) = f\in H^{-1}(\Omega) \] weakly converge in \(H^ 1_ 0(\Omega)\) to the solution \(u_ 0\) of the problem \[ A_ 0u_ 0+c_ 0(x)g(x,u_ 0)=f \] and \[ A_{\epsilon} u_{\epsilon}\to A_ 0u_ 0\text{ in } H^{-1}(\Omega), \] \[ c_{\epsilon}(x)g(x,u_{\epsilon})\to c_ 0(x)g(x,u_ 0)\text{ in } D'(\Omega). \]
scientific article; zbMATH DE number 4019580