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Non-existence results for semilinear cooperative elliptic systems via moving spheres. (English) Zbl 0962.35054

Nonexistence results for positive (nonnegative) solutions for cooperative elliptic systems of the (vector) form $-{\Delta }u=f\left(u\right)$ in $D$, $u=0$ on $\partial D$ are proved via the method of moving spheres. This constitutes a variant of the celebrated moving plane method due to Alexandrov and applied by Serrin, Gidas-Nirenberg and many others to get symmetry results for positive solutions. This approach of moving spheres was used by McCuan and Padilla for obtaining symmetry results, too. The moving sphere method, in the authors’ claim, unifies and simplifies previous work, even for systems. The method of moving planes raises compactness problems when dealing with unbounded domains, a difficulty which can be overcome in this way.

Theorem 1 states that if $D\subset {ℝ}^{n}$ $\left(n\ge 3\right)$ is a bounded starshaped domain and $f$ is locally Lipschitz and supercritical, there is no positive solution. Some singularities in the $x$ variable can be allowed. A similar result (Theorem 2) is proved if $D$ is starshaped with respect to infinity and $f$ is subcritical (again in a suitable sense). This gives as corollaries nonexistence results by Gidas and Spruck on ${ℝ}^{n}$ and the half-space, and also for “curved” half-spaces. An interesting monotonicity result (Theorem 3) is very instrumental here. Theorem 4 is an interesting corollary for power nonlinearities and Theorem 5 says that nonnegative solutions to $-{\Delta }u=f\left(u\right)$ on ${ℝ}_{+}^{n}$ with $u=0$ on $\partial {ℝ}_{+}^{n}$ for some subcritical $f$’s depend only on ${x}_{1}$ and are increasing. Some applications to singular problems are also included. In particular, proofs use many subtle comparison arguments and variants of maximum principles.

##### MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35B50 Maximum principles (PDE) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35J65 Nonlinear boundary value problems for linear elliptic equations