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semilinear elliptic problem

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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Existence results for semilinear problems in the two-dimensional hyperbolic space involving critical growth
Part of
Debdip Ganguly, Debabrata Karmakar
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 147 / Issue 1 / February 2017

Published online by Cambridge University Press:

06 January 2017, pp. 141-176

Print publication:

February 2017
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We consider semilinear elliptic problems on two-dimensional hyperbolic space. A model problem of our study is

where H 1(𝔹2) denotes the Sobolev space on the disc model of the hyperbolic space and f(x, t) denotes the function of critical growth in dimension 2. We first establish the Palais–Smale (PS) condition for the functional corresponding to the above equation, and using the PS condition we obtain existence of solutions. In addition, using a concentration argument, we also explore existence of infinitely many sign-changing solutions.