## On the existence of a positive solution of semilinear elliptic equations in unbounded domains.(English)Zbl 0883.35045

The authors prove the existence of solutions to $-\Delta u+\lambda_0u= b(x)u^p\quad\text{in }\Omega\subset\mathbb{R}^n,\quad u\in H^1_0(\Omega),\quad u>0\quad\text{in }\Omega,$ where $$\lambda_0>0$$, $$1<p<(n+2)/(n-2)$$ if $$n\geq 3$$ ($$1<p<\infty$$ if $$n=2$$), and $$\Omega=\overline O^c$$, $$O$$ being a smooth bounded open set. The weight function $$b\in C_0(\mathbb{R}^n)$$ is assumed to be positive, $$b\to b^\infty$$ as $$|x|\to\infty$$, and $$b(x)\geq b^\infty- c_0\exp(-\delta|x|)|x|^{-(n-1)/2}$$ in a neighborhood of infinity with constants $$c_0\geq 0$$, $$\delta>0$$. Nonnegative solutions of this problem are nonnegative critical points of $I(v)\equiv\int_\Omega {1\over 2}|\nabla v|^2+ {1\over 2}\lambda_0v^2- {1\over p+1} b|v|^{p+1}dx,\quad v\in H^1_0(\Omega).$ Such points correspond to nonnegative critical points of $J(v)\equiv \sup_{\lambda\geq 0} I(\lambda v),\quad v\in\Biggl\{w\in H^1_0(\Omega)|\int_\Omega|\nabla w|^2+\lambda_0 w^2dx= 1\Biggr\}.$ Both functionals do not satisfy a Palais-Smale condition. However, the authors are able to characterize the sequences violating the Palais-Smale condition. Assuming that the above problem has no solution, this characterization enables them to prove a deformation lemma for the level sets of $$J$$. Together with a topological argument, this leads to a contradiction.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations

### Keywords:

exterior domain; Palais-Smale condition; deformation lemma
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### References:

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