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)


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
Full Text: DOI


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