Global Solutions

In the previous two chapters, we studied the semilinear Schrödinger equation − Δ u + V ( x ) u = f ( x , u ) , u ∈ H 1 ( ℝ n ) , $$\displaystyle - \Delta u + V(x) u = f(x,u), \quad u \in H^1({\mathbb R^n}), $$ where V (x) is a given potential. We needed the linear operator − Δu + V (x)u to have a nonempty resolvent. To achieve this, we assumed that V (x) was periodic in x. This forced us to assume the same for f(x, u), and we had to deal with several restrictions in our methods. In this chapter we study the equation without making any periodicity assumptions on the potential or on the nonlinear term. But we must be assured that the linear operator has nonempty resolvent. To accomplish this, we make assumptions on V (x) which guarantee that the essential spectrum of − Δu + V (x)u is the same as that of − Δu. In other words, our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in [0, ∞) being the absolutely continuous part of the spectrum. There may be no negative eigenvalues, a finite number of negative eigenvalues, or an infinite number of negative eigenvalues. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.