Strong stability results for nonlinear elliptic equations with respect to very singular perturbation of the data (Q2732616)

The motivation of the present work is the study of nonlinear elliptic equations of the form \(-\Delta u+h(u)|\nabla u|^2=f\), in \(\Omega \setminus K\), where \(f\) belongs to \(L^\infty (\Omega)\), \(K\) is a compact subset of \(\Omega\) with zero capacity and \(h\) is a continuous function such that: \(h(s)s\geq \gamma >1\), for every \(s\geq s_0>0\).NEWLINENEWLINE\textit{H. Brézis} and \textit{L. Nirenberg} [Topol. Methods Nonlinear Anal. 9, No. 2, 201--219 (1997; Zbl 0905.35027)] proved that if \(u\) is a smooth solution to this problem in \(\Omega \subset K\), then it has to be a smooth function in the whole \(\Omega \). This leads to a non-existence result, when replacing \(f\) by \(f+\mu\), where \(\mu\) is a measure with support in a compact set with zero capacity. Then, Brézis raised the conjecture concerning the behavior of the solution \(u_n\) of the preceding equation, when the second member is \(f_n\) (\(f_n\) in \(L^\infty(\Omega)\)) and \((f_n)_n\) converges to \(f\) in a weak sense, namely, NEWLINE\[NEWLINE\int_{\Omega \setminus I(K)}|f_n-f|\,dxNEWLINE\]NEWLINE tends to 0, for every neighborhood \(I(K)\) of \(K\).NEWLINENEWLINEThe authors solve this conjecture in a generalized way, considering the equation NEWLINE\[NEWLINE-\text{div}\,a(x,u,\nabla u)+H(x,u,\nabla u)=f, \quad \text{in}\;\Omega,NEWLINE\]NEWLINE with homogeneous Dirichlet boundary conditions on \(\partial \Omega\). The Carathéodory function \(a\) is assumed to have a \((p-1)\)-growth with respect to its last variable: \(|a(x,s,\xi)|\leq\beta|\xi|^{p-1}\) and \(a(x,s,\xi )\cdot\xi\geq \alpha|\xi|^p\), for \(\alpha \) and \(\beta \) positive; \(a\) is also assumed to be monotone with respect to this last variable. The Carathéodory function \(H\) is assumed to have a \(p\)-growth with respect to its last variable: \(g(s)|\xi|^p\leq H(x,s,\xi )\leq \theta (x)+b(s)|\xi|^p\), where \(g\) satisfies some growth conditions.NEWLINENEWLINEThe main result asserts that if \((f_n)_n\) (with \(f_n\) nonnegative) converges in the preceding weak sense to \(f\) and if \(u_n\) is the solution of the generalized nonlinear equation in \(W_0^{1,p}(\Omega )\cap L^\infty (\Omega )\) , then \((u_n)_n\) converges almost everywhere in \(\Omega \) to a generalized solution of the preceding generalized nonlinear equation with second member equal to \(f\). The gradient of this generalized solution \(u\) is indeed defined by means of the truncation operators \(T_k\) (\(T_k(s)=\max (-s,\min (k,s))\)), through \(\nabla u\chi_{{|u|\leq k}}=\nabla T_k(u)\), almost everywhere in \(\Omega \), and for every \(k\).NEWLINENEWLINEThe proof of this interesting result relies heavily on these truncation operators in order to obtain estimates on the solution \(u_n\), and on the nonnegativity of \(u_n\).NEWLINENEWLINEThe authors conclude their work by proving that their result is sharp with respect to some growth condition on \(g\). Indeed, they build a counterexample.