We prove an abstract result of existence of "good" generalized subsolutions for convex operators. Its application to semilinear elliptic equations leads to an extension of results by P.B-M.Pierre concerning a criterion for the existence of solutions to a semilinear elliptic or parabolic equation with a convex nonlinearity. We apply this result to the model problem $ -\Delta u = a |\nabla u|^p+ b|u|^q+f $ with Dirichlet boundary conditions where $ a,b>0 $, $ p,q>1 $. No other condition is made on $ p $ and $ q $.

中文翻译：

---

半线性椭圆型方程解存在性准则的一般化

我们证明了凸算子“好”广义子解的存在的抽象结果。它在半线性椭圆方程上的应用导致PB-M.Pierre扩展了关于具有凸非线性的半线性椭圆或抛物方程的解的存在性准则。我们将此结果应用于具有Dirichlet边界条件的模型问题$-\ Delta u = a | \ nabla u | ^ p + b | u | ^ q + f $，其中$ a，b> 0 $，$ p，q> 1 $。$ p $和$ q $没有其他条件。