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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
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References:

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