Single Peak Solutions for a Schrödinger Equation with Variable Exponent

We study the following Schrödinger equation with variable exponent

$$ - \Delta u + u = {u^{p + \epsilon a(x)}},\,\,\,u > 0\,\,{\rm{in}}\,\,{\mathbb{R}^N},$$

where \(\epsilon > 0,\,\,1 < p < {{N + 2} \over {N - 2}},\,\,a(x) \in {C^1}({\mathbb{R}^N}) \cap {L^\infty }({\mathbb{R}^N}),\,\,N \ge 3\) Under certain assumptions on a vector field related to a(x), we use the Lyapunov–Schmidt reduction to show the existence of single peak solutions to the above problem. We also obtain local uniqueness and exact multiplicity results for this problem by the Pohozaev type identity.