论文标题
$ l^\ infty $ a-priori估计，用于carathéodory非线性的亚临界半椭圆方程
$L^\infty$ a-priori estimates for subcritical semilinear elliptic equations with a Carathéodory nonlinearity
论文作者
论文摘要
我们提出了新的$ l^\ infty $ a先验估计，以估计有限域中一类较大的亚临界椭圆方程的弱解决方案。在解决方案的迹象上没有假设，都不需要两个非线性。该方法基于将椭圆规律与Gagliardo-Nirenberg或Caffarelli-Kohn-Nirenberg插值不平等相结合。 让我们考虑一个半线性边界价值问题$-ΔU= f（x，u），$ in $ω，带有dirichlet边界条件的$，其中$ω\ subset \ mathbb {r}^n $，带有$ n> 2，$是一个有界的平稳域，$ f $是$ f $ $ f $ subectialCalliticalCarathéodorycarathéodorynor -nor -lel -lerneareity。我们提供了$ l^\ infty $ a的先验估计，以$ l^{2^*} $ - norm，其中$ 2^*= \ frac {2n} {n-2} {n-2} \ $是关键的sobolev endonent。 通过亚临界非线性，我们的意思是，例如，$ | f（x，s）| \ le | x |^{ - μ} \ \，\ tilde {f}（s），$ where $μ\ in（0,2），$和$ \ tilde和$ \ tilde和$ \ tilde {f}（f}（f}（s）/| s） \ infty $，这里$ 2^*_μ：= \ frac {2（n-μ）} {n-2} $是关键的sobolev-hardy指数。我们的非线性包括非电力非线性。特别是我们证明，当$ f（x，s）= | x |^{ - μ} \，\ frac {| s |^{2^*_μ-μ-2} s} s} {\ big [\ big [\ log（e+| s |） $ c_ \ varepsilon> 0 $使得对于任何解决方案$ u \ in H^1_0（ω）$，以下内容保留$$ \ big [\ log \ big（e+\ | | | _ {\ | _ {\ infty} \ big）\ big] \ big] \ big] \ big（1+ \ | U \ | _ {2^*} \ big）^{\，（2^*_μ-2）（1+ \ varepsilon）} \，。 $$
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical elliptic equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in combining elliptic regularity with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a semilinear boundary value problem $ -Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $N> 2,$ is a bounded smooth domain, and $f$ is a subcritical Carathéodory non-linearity. We provide $L^\infty$ a priori estimates for weak solutions, in terms of their $L^{2^*}$-norm, where $2^*=\frac{2N}{N-2}\ $ is the critical Sobolev exponent. By a subcritical non-linearity we mean, for instance, $|f(x,s)|\le |x|^{-μ}\, \tilde{f}(s),$ where $μ\in(0,2),$ and $\tilde{f}(s)/|s|^{2_μ^*-1}\to 0$ as $|s|\to \infty$, here $2^*_μ:=\frac{2(N-μ)}{N-2}$ is the critical Sobolev-Hardy exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when $f(x,s)=|x|^{-μ}\,\frac{|s|^{2^*_μ-2}s}{\big[\log(e+|s|)\big]^β}\,,$ with $μ\in[1,2),$ then, for any $\varepsilon>0$ there exists a constant $C_\varepsilon>0$ such that for any solution $u\in H^1_0(Ω)$, the following holds $$ \Big[\log\big(e+\|u\|_{\infty}\big)\Big]^β\le C _\varepsilon \, \Big(1+\|u\|_{2^*}\Big)^{\, (2^*_μ-2)(1+\varepsilon)}\, . $$
