## 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

### Keywords:

positive solutions; Dirichlet problem; superlinearity

Zbl 0583.00018