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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
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