Existence results for semilinear problems in the two-dimensional hyperbolic space involving critical growth

Abstract

We consider semilinear elliptic problems on two-dimensional hyperbolic space. A model problem of our study iswhere H1(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.