Dual ground state solutions for the critical nonlinear Helmholtz equation

Abstract

Using a dual variational approach we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation $$ - \Delta u - k^{2}u = Q(x)|u|^{2^{\ast} - 2}u, \quad u \in W^{2,2^{\ast}}(\mathbb{R}^{N}) $$ for $N\geq 4$, where $2^{\ast} := \frac{2N}{N-2}$. The coefficient $Q \in L^{\infty}(\mathbb{R}^{N})\setminus\{0\}$ 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, i.e., 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 $N=3$.