We consider the nonlinear elliptic equation with quadratic convection \( -\Delta u + g(u) |\nabla u|^2=\lambda f(u) \) in a smooth bounded domain \( \Omega \subset {\mathbb {R}}^N \) (\( N \ge 3\)) with zero Dirichlet boundary condition. Here, \( \lambda \) is a positive parameter, \( f:[0, \infty ):(0\infty ) \) is a strictly increasing function of class \(C^1\), and g is a continuous positive decreasing function in \( (0, \infty ) \) and integrable in a neighborhood of zero. Under natural hypotheses on the nonlinearities f and g, we provide some new regularity results for the extremal solution \(u^*\). A feature of this paper is that our main contributions require neither the convexity (even at infinity) of the function \( h(t)=f(t)e^{-\int _0^t g(s)ds}\), nor that the functions \( gh/h'\) or \( h'' h/h'^2\) admit a limit at infinity.

我们考虑在光滑有界域\（\ Omega \ subset {\ mathbb中，）具有二次对流\（-\ Delta u + g（u）| \ nabla u | ^ 2 = \ lambda f（u）\）的非线性椭圆方程{R}} ^ N \）（\（N \ ge 3 \）），Dirichlet边界条件为零。在这里，\（\ lambda \）是一个正参数，\（f：[0，\ infty）:( 0 \ infty）\）是类\（C ^ 1 \）的严格增加的函数，g是一个\（（0，\ infty）\）中的连续正递减函数，并且可在零附近积分。在非线性f和g的自然假设下，我们为极值解\（u ^ * \）提供了一些新的正则结果。本文的一个特点是，我们的主要贡献不需要函数\（h（t）= f（t）e ^ {-\ int _0 ^ tg（s）ds} \）的凸性（甚至是无穷大），函数\（gh / h'\）或\（h''h / h'^ 2 \）也不允许无穷大。