Published online by Cambridge University Press: 30 January 2019
Using a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation
$$-\Delta u-k^2u = Q(x) \vert u \vert ^{2^*-2}u,\quad u\in W^{2,2^*}({\open R}^{N})$$
$Q \in L^{\infty }({\open R}^{N}){\setminus }\{0\}$ is assumed to be nonnegative, asymptotically periodic and to satisfy a flatness condition at one of its maximum points. The solutions obtained are so-called dual ground states, that is, solutions arising from critical points of the dual functional with the property of having minimal energy among all nontrivial critical points. Moreover, we show that no dual ground state exists for N = 3.
Present address: School of Engineering, University of Applied Sciences of Western Switzerland, Route du Rawil 47, 1950 Sion, Switzerland ([email protected])