Existence for asymptotically coercive nonlinear elliptic equations in Hilbert spaces
Ronald I. Becker · Hokkaido Mathematical Journal · 1993
We consider equations of the form (L-Q(x))x+e=0 where 1. L is a self-adjoint (abstract) elliptic operator with domain in H_{1}\subseteq H ( H a Hilbert space); 2. For each x\in H , Q(x) is a bounded self-adjoint linear operator on H ; 3. e\in H (see Section 2 for a definition of (abstract) elliptic).Many results in the literature deal with the case where L is a differential operator and, if \lambda_{n} and \lambda_{n+1} are successive eigenvalues of L, then we have for all x of sufficiently large norm \lambda_{n}I\leq Q(x)\leq\lambda_{n+1}I(where inequality is in the usual partial order on the self-adjoint opera- tors).This is not sufficient to guarantee existence, since Q may interact with the eigenvectors corresponding to the two eigenvalues.Suitable sufficient conditions are that for all x of sufficiently large normIn this paper, we will relax the condition at the lower eigenvalue to Q(x) \geq\lambda_{n}I for all x of sufficiently large norm and an asymptotic coerciveness- type condition of the form \lim_{marrow}\inf_{\infty}((Q(x_{m})-\lambda_{n})x_{m}-e, T_{n}x_{m}))>0 for all sequences \{x_{m}\} tending asymptotically to eigenspace of \lambda_{n} (see Section 3 for a precise definition).1980