Equations Elliptiques Semi Lineaires dans des Domaines Non Bornes de $\R^{N}$

Khodja Brahim

Abstract: Let $f$ be a locally lipschitz continuous function verifying $f(0)=0$. We consider the problem $$ \cases{-\Delta u+f(u)=0& $u\in H^{2}(\Omega)\cap L^{\infty}(\Omega)$,\cr Bu=0& on $\Gamma\virg$}\leqno({\rm P}) $$ with $Bu=u$ (Dirichlet's condition) or $Bu=u_{n}$ (Neumann's condition). We show that if $$ \Omega=\R^{2}\times G\rmdu{where} \hbox{$G$ is an open set of $\R^{N}$} $$ or $$ \Omega\rmdu{is an open set of}\R^{N} (N\ge2), $$ verifying $\exists\,\chi\in\R^{N}$, $\chi$ constant $\langle n,\chi\rangle>0$ on $\Gamma$ and $F(u)\ge0$.
The (P) problem admits only the zero solution.\Prgrf We have the same result if $$ \Omega=\Bigl\{(x_{1},x_{2},...,x_{N})\in\R^{N}\,| x_{1}>\Phi(x_{2},...,x_{N})\Bigr\} $$ with $$ \Phi(x_{2},...,x_{N})=\sigma_{2}|x_{2}|^{\alpha}+...+ \sigma_{N}|x_{N}|^{\alpha}, \alpha>1\rmdd{and}\sigma_{i}\in\R $$ and $$ F(u)\le c(\alpha)\,u\,f(u). $$

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