## Positive solutions of semilinear equations in cones.(English)Zbl 0766.35014

The paper deals with the problem $$\Delta u+| x|^ \nu u^{(n+2+2\nu)/(n-2)}=0$$ in a cone $${\mathcal C}\subset\mathbb{R}^ n$$, $$n\geq 3$$, $$u>0$$ in $${\mathcal C}$$, $$u=0$$ on $$\partial{\mathcal C}$$ and $$u(x)=o(| x|^{2-n})$$ as $$| x|\to\infty$$. This problem is the critical case which was left open in the paper on Emden equations in cones [Arch. Ration. Mech. Anal. 112, No. 4, 319-338 (1990; Zbl 0727.35051)] by C. Bandle and M. Essén. It is shown that the existence depends on the sign of $$\nu$$. The author uses variational methods to prove the existence for $$\nu\in(-2,0)$$. Subtle arguments are needed because of lack of compactness. In the case $$\nu=0$$ the existence is established for special cones. They provide examples of simply connected domains for which the Emden equations with critical exponent possess solutions.
Reviewer: C.Bandle (Basel)

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35J25 Boundary value problems for second-order elliptic equations 35J20 Variational methods for second-order elliptic equations

Zbl 0727.35051
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### References:

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