We consider a semilinear boundary value problem \( -\Delta u= f(x,u),\) in \(\Omega ,\) with Dirichlet boundary conditions, where \(\Omega \subset {\mathbb {R}}^N \) with \(N> 2,\) is a bounded smooth domain, and f is a Carathéodory function, superlinear and subcritical at infinity. We provide \(L^\infty (\Omega )\) a priori estimates for weak solutions in terms of their \(L^{2^*}(\Omega )\)-norm, where \(2^*=\frac{2N}{N-2}\ \) is the critical Sobolev exponent. In particular, our results also apply to \(f(x,s)=a(x)\,\frac{|s|^{2^*_{N/r}-2}s}{\big [\log (e+|s|)\big ]^\beta }\,\), where \(a\in L^r(\Omega )\) with \(N/2<r\le \infty \), and \(2_{N/r}^*:=2^*\left( 1-\frac{1}{r}\right) \). Assume \(N/2<r\le N\). We show that for any \(\varepsilon >0\) there exists a constant \(C_\varepsilon >0\) such that for any solution \(u\in H^1_0(\Omega )\), the following holds:

To establish our results, we do not assume any restrictions on the sign of the solutions, or on the non-linearity. Our approach is based on Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg interpolation inequalities. Finally, we state sufficient conditions for having \(H_0^1(\Omega )\) uniform a priori bounds for non-negative solutions, so finally we provide suficient conditions for having \(L^\infty (\Omega )\) uniform a priori bounds, which holds roughly speaking for superlinear and subcritical non-linearities.

我们考虑一个半线性边界值问题\( -\Delta u= f(x,u),\) in \(\Omega ,\)具有 Dirichlet 边界条件，其中\(\Omega \subset {\mathbb {R}} ^N \)其中\(N> 2,\)是有界平滑域，f是 Carathéodory 函数，在无穷远处超线性和亚临界。我们根据\(L^{2^*}(\Omega )\) -范数提供\(L^\infty (\Omega )\)弱解的先验估计，其中\(2^*=\ frac{2N}{N-2}\ \)是临界 Sobolev 指数。特别地，我们的结果也适用于\(f(x,s)=a(x)\,\frac{|s|^{2^*_{N/r}-2}s}{\big [\ log (e+|s|)\big ]^\beta }\,\)，其中\(a\in L^r(\Omega )\)与\(N/2<r\le \infty \)和\(2_{N/r}^*:=2^*\left( 1- \frac{1}{r}\右) \)。假设\(N/2<r\le N\)。我们证明对于任何\(\varepsilon >0\)存在一个常数\(C_\varepsilon >0\)这样对于任何解\(u\in H^1_0(\Omega )\)，以下成立：

为了确定我们的结果，我们不对解的符号或非线性假设任何限制。我们的方法基于 Gagliardo–Nirenberg 和 Caffarelli–Kohn–Nirenberg 插值不等式。最后，我们陈述了使\(H_0^1(\Omega )\)均匀非负解的先验边界的充分条件，所以最后我们提供了使\(L^\infty (\Omega )\)均匀的充分条件先验边界，粗略地适用于超线性和亚临界非线性。