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Existence theorems for superlinear elliptic Dirichlet problems in exterior domains. (English) Zbl 0596.35048

Nonlinear functional analysis and its applications, Proc. Summer Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45/1, 271-282 (1986).
The authors prove the existence of positive \(W_ 0^{1,2}(\Omega)\) solutions to the Dirichlet problem \[ \Delta u-u+f(u)=0\quad\text{in} \;\Omega,\quad u=0\quad\text{on}\;\partial \Omega, \] for two nonradially-symmetric type domains \(\Omega\) in the exterior of a compact domain. In the first case \(\Omega\) is invariant under a group of orthogonal transformations of \(\mathbb R^ 3\) having only the origin as a common fixed point and in the second case \(\partial \Omega\) is contained in a sufficiently thin ring about the sphere \(| x| =R\). The results are stated in \(\mathbb R^ 3\) for simplicity. Here \(f\in C^ 1(\mathbb R)\) satisfies \(f(0)=0\), \(f(u)>0\) for \(u>0\) and the superlinearity condition \(u f(u)>(2+\varepsilon)F(u)\) for some \(\varepsilon >0\) and all \(u>0\) and where \(F(u)=\int^{u}_{0}f(t)\,dt.\)
The solution is obtained in the first case as a limit of solutions to approximating problems on bounded domains which minimize the associated variational problem. In the second case, perturbation methods and analysis of a related eigenvalue problem are used.
[For the entire collection see Zbl 0583.00018.]

MSC:

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

Citations:

Zbl 0583.00018