# Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

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We prove a general alternate circular summation formula of theta functions, which yields a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulas for (q; q)2nâˆž.

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We consider a problem with local shift conditions on a segment of the degeneration line and with shifts in interior characteristics of the equation. The uniqueness of the solution is proved with the use of the extremum principle. To prove the existence of a solution, we apply the theory of...