论文标题
$ l^\ infty $ a-priori估计值$ p $ - laplacian方程，具有carathéodory非线性
$L^\infty$ a-priori estimates for subcritical $p$-laplacian 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，以估计有界域中一类宽类$ p $ laplacian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。此方法基于椭圆规律性的$ p $ -laplacian与Gagliardo-Nirenberg或Caffarelli-Kohn-Nirenberg插值不平等相结合。 让我们考虑一个quasilinear边界价值问题$-Δ_Pu = f（x，u），$ω，$ $ω，带有dirichlet边界条件的$，其中$ω\ subset \ subset \ mathbb {r}^n $，带有$ p <n，$是有界的平稳域严格凸出，$ $ f $是$ f $carathéod的carathéodedecarathéodorycarathéodory。我们提供了$ l^\ infty $ a的弱解决方案的先验估计，就其$ l^{p^*} $ - norm而言，其中$ p^*= \ frac {np} {n-p} {n-p} \ $是关键的sobolev endent。 通过亚临界非线性，我们的意思是，例如，$ | f（x，s）| \ le | x |^{ - μ} \，\，\ tilde {f}（s），$ where $μ\ in（0，p），$和$ \ tilde，$＆\ tilde {f}（f}（f}（s）/| s） \ infty $，这里$ p^*_μ：= \ frac {p（n-μ）} {n-p} $是关键的sobolev-hardy指数。我们的非线性包括非电力非线性。 特别是我们证明，当$ f（x，s）= | x |^{ - μ} \，\ frac {| s | s |^{p^*_μ-μ-2} s} {\ big [\ big [\ log（e+| s |） $ c_ \ varepsilon> 0 $使得对于任何解决方案$ u \在h^1_0（ω）$中，以下内容保留$$ \ big [\ log \ big（e+\ | U \ | | _ {\ infty} \ big）\ big] \ big] \ big（1+ \ | U \ | _ {p^*} \ big）^{\，（p^*_μ-p）（1+ \ varepsilon）} \，，$ $，其中$ c_ \ varepsilon $独立于解决方案$ u $。
We present new $L^\infty$ a priori estimates for weak solutions of a wide class of subcritical $p$-laplacian equations in bounded domains. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in elliptic regularity for the $p$-laplacian combined either with Gagliardo-Nirenberg or Caffarelli-Kohn-Nirenberg interpolation inequalities. Let us consider a quasilinear boundary value problem $ -Δ_p u= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $Ω\subset \mathbb{R}^N $, with $p<N,$ is a bounded smooth domain strictly convex, and $f$ is a subcritical Carathéodory non-linearity. We provide $L^\infty$ a priori estimates for weak solutions, in terms of their $L^{p^*}$-norm, where $p^*= \frac{Np}{N-p}\ $ is the critical Sobolev exponent. By a subcritical non-linearity we mean, for instance, $|f(x,s)|\le |x|^{-μ}\, \tilde{f}(s),$ where $μ\in(0,p),$ and $\tilde{f}(s)/|s|^{p_μ^*-1}\to 0$ as $|s|\to \infty$, here $p^*_μ:=\frac{p(N-μ)}{N-p}$ is the critical Sobolev-Hardy exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when $f(x,s)=|x|^{-μ}\,\frac{|s|^{p^*_μ-2}s}{\big[\log(e+|s|)\big]^α}\,,$ with $μ\in[1,p),$ then, for any $\varepsilon>0$ there exists a constant $C_\varepsilon>0$ such that for any solution $u\in H^1_0(Ω)$, the following holds $$ \Big[\log\big(e+\|u\|_{\infty}\big)\Big]^α\le C_\varepsilon \, \Big(1+\|u\|_{p^*}\Big)^{\, (p^*_μ-p)(1+\varepsilon)}\, , $$ where $C_\varepsilon$ is independent of the solution $u$.
