Eigenvalue asymptotics and a nonlinear Schrödinger equation (Q1261913)

Let \(S\) be the unit sphere in \(L^ 2(\mathbb{R}^ N)\), \(N \geq 3\) and \(r>0\). The authors show the existence of infinitely many eigenvalues \(\lambda_ n\) of the problem \[ -\Delta u+qu+f(x,u)=\lambda u \text{ in }\mathbb{R}^ N,\qquad u\in W^{1,2}(\mathbb{R}^ N)\cap rS. \] The asymptotic behavior of \(\lambda_ n\) as \(n \to \infty\) is also studied. The function \(q\) is assumed to be bounded from below, \(q(x) \to \infty\) as \(| x | \to \infty\), the perturbation term \(f\) is assumed to be a continuous function, \(f(x,-s)=-f(x,s)\), \(f(x,s)s \geq 0\) and \(| f(x,s)| \leq a | s |^{p-1}+b(x)\), where \(b \in L^ 2(\mathbb{R}^ N)\) and \(p<2+ 4/N\). The proofs use the Lyusternik-Schnirelman theory and they rely on the compactness of the embedding of the weighted Sobolev space \(W_ q^{1,2} (\mathbb{R}^ N)=\{u;\int_{\mathbb{R}^ N}(| \nabla u |^ 2+qu^ 2)dx<\infty\}\) in \(L^ p(\mathbb{R}^ N)\).