Sharp Sobolev inequalities in critical dimensions. (English) Zbl 1195.35132
From the introduction: Let \(K\in\mathbb N\) and \(\Omega\subset \mathbb R^N\) \((N> 2K+1)\) be a regular bounded domain in \(\mathbb R^N\). We consider the semilinear polyharmonic problem
\[ (-\Delta)^Ku=\lambda u+ | u|^{s-2}u\;\text{in}\;\Omega \tag{1} \]
where \(s:=\frac{2N}{N-2K}\) denotes the critical Sobolev exponent. Generalizing results of Brezis-Nirenberg, Pucci-Serrin and Gazzola-Grunau the following sharp estimate is proved.
Theorem: Let \(2k+1\leq n\leq 4k-1\). Then, for any \(1\leq q<n/(n-2k)\), there exists a constant \(c>0\) such that \[ S_k\| f\|_{L^s(\Omega)}^2+c\| f\|_{L^q(\Omega)}^2\leq \| f\|_{k,2,\Omega}^2\quad (s=2n/(n-2k))\tag{1} \] for every \(f\in H_0^k(\Omega)\).
\[ (-\Delta)^Ku=\lambda u+ | u|^{s-2}u\;\text{in}\;\Omega \tag{1} \]
where \(s:=\frac{2N}{N-2K}\) denotes the critical Sobolev exponent. Generalizing results of Brezis-Nirenberg, Pucci-Serrin and Gazzola-Grunau the following sharp estimate is proved.
Theorem: Let \(2k+1\leq n\leq 4k-1\). Then, for any \(1\leq q<n/(n-2k)\), there exists a constant \(c>0\) such that \[ S_k\| f\|_{L^s(\Omega)}^2+c\| f\|_{L^q(\Omega)}^2\leq \| f\|_{k,2,\Omega}^2\quad (s=2n/(n-2k))\tag{1} \] for every \(f\in H_0^k(\Omega)\).
MSC:
| 35J35 | Variational methods for higher-order elliptic equations |
| 35B33 | Critical exponents in context of PDEs |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
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