Nonresonance theory for semilinear operator equations under regularity conditions.(English)Zbl 1169.47053

Given a Hilbert space $$H$$ with inner product $$\langle\cdot,\cdot\rangle_H$$ and a selfadjoint positive operator $$A:D(A)\to H$$ in $$H$$ with compact resolvent, the energy space $$H_A$$ of $$A$$ is defined as $$H_A:=A^{-1/2}(H)$$, with inner product $$\langle u,v\rangle_{H_A}:=\langle A^{1/2}u,A^{1/2}v\rangle_H$$. Suppose that $$Y$$ is another Hilbert space, that $$S\in\mathcal{L}(H_A,Y)$$ satisfies $$\|S\|_{\mathcal{L}(H_A,Y)}\leq 1$$, and that $$F: H\times Y\to H$$ is a continuous map. Then the nonlinear operator equation
$Au=cu+F(u,Su),\qquad u\in H_A,\tag{1}$
is considered in the nonresonant case $$c\notin\sigma(A)$$. Here, one looks for weak solutions of (1), i.e., elements $$u\in H_A$$ such that $$\langle u,v\rangle_{H_A}=\langle cu+F(u,Su),v\rangle_H$$ for all $$v\in H_A$$.
Using Banach’s fixed point theorem and Leray-Schauder theory, various existence theorems are given for (1). The conditions on $$F$$ amount to at most linear growth in $$u$$ and $$v$$, and in some instances include (partial) global Lipschitz conditions on $$F$$. The growth and Lipschitz constants appearing here are of the order $$1/\|(A-c)^{-1}\|$$. In some cases, the existence of a closed subspace $$Z$$ of $$Y$$ is assumed that embeds compactly and satisfies $$D_A\subseteq S^{-1}(Z)$$. The authors call this assumption a regularity condition.
One application is the following. Let $$\Omega\subseteq\mathbb{R}^N$$ denote a smoothly bounded domain. Let $$A$$ denote $$-\Delta$$ on $$\Omega$$ with respect to Dirichlet boundary conditions. Hence $$E:=L^2(\Omega)$$ and $$H_A:=H^1_0(\Omega)$$. Suppose that $$c$$ is not an eigenvalue of $$A$$. Let $$Y:=L^2(\Omega,\mathbb{R}^N)$$, $$S:=\nabla$$, and $$Z:=H^1(\Omega,\mathbb{R}^N)$$. Then $$D(A)=H^1_0(\Omega)\cap H^2(\Omega)\subseteq S^{-1}(Z)=H^2(\Omega)$$.
Suppose that $$f:\mathbb{R}\times\mathbb{R}^N\to\mathbb{R}$$ induces a continuous superposition operator $$F: H\times Y\to H$$ that satisfies
$\|F(u,v)\|_{H}\leq a\|u\|_{H}+b\|v\|_{Y}+h$
for some small enough $$a,b>0$$ and some $$h>0$$. Then there is at least one solution $$u\in H^1_0(\Omega)\cap H^2(\Omega)$$ of
$-\Delta u=cu+f(u,\nabla u).$
The main point seems to be that $$F$$ is only bounded linearly in $$u$$ and $$v$$, not necessarily sublinearly.

MSC:

 47J25 Iterative procedures involving nonlinear operators 35J65 Nonlinear boundary value problems for linear elliptic equations