Semilinear problems with a non-symmetric linear part having an infinite dimensional kernel (Q1769304)

This article deals with equations of the form \[ Lu = N(u) + h, \] where \(L: D(L) (\subseteq H) \to H\) is a densely defined unbounded closed linear operator on a real separable Hilbert space \(H\), \(N: H \to H\) is a nonlinear operator on \(H\), and \(h \in H\). The authors consider the case when \(\text{Ker}\,L\neq\text{Ker}\,L^*\), \(\dim\text{Ker}\,L = \dim\text{Ker}\,L^*\leq\infty\), and \(\| Lu\| ^2 \geq \rho \, \langle Lu,Ju \rangle\) \ (\(J\) is a linear homeomorphism of \(H\), \(J(\text{Ker}\,L) = \text{Ker}\,L^*\)); as for \(N\), the authors assume that it is bounded and demicontinuous and that it satisfies one of the following properties: (a) \(T\)-monotonicity (namely, \(\langle N(u) - N(v),T(u - v) \rangle \geq 0\) for all \(u, v \in H\)); (b) being of class \((S_+)_T\) (namely, if \(v_j \in \text{Ker}\,T\), \(z_j \in (\text{Ker}\,T)^\bot\), \(u_j\rightharpoonup u\), \(v_j \to v\), and \(\limsup \langle F(u_j),T(u_j - u) \rangle \leq 0\), then \(u_j = v_j + z_j \to u + v\)); (c) \(T\)-pseudomonotonicity (namely, if \(v_j \in \text{Ker}\,T\), \(z_j \in (\text{Ker}\,T)^\bot\), \(u_j \rightharpoonup u\), \(v_j \to v\), and \(\limsup\langle F(u_j),T(u_j - u) \rangle \leq 0\), then \(F(u_j) \rightharpoonup F(u)\) and \(\langle F(u_j),T(u_j - u) \rangle \to 0\), where \(u_j = v_j + z_j\), \(u = v +z\)); (d) \(T\)-quasimonotonicity (namely, if \(v_j \in \text{Ker}\,T\), \(z_j \in (\text{Ker}\,T)^\bot\), \(u_j \rightharpoonup u\), \(v_j \to v\), then \(\limsup\langle F(u_j),T(u_j - u) \rangle \geq 0\), where again \(u_j = v_j + z_j\), \(u = v +z\)); in all these definitions, \(T = JP\), where \(P\) is the orthogonal projection on \(\text{Ker}\;L\). Moreover, the authors assume that (the non-resonance case) \[ \left\| N(u) - \frac\rho2 Ju\right\| \leq \mu\| Ju\| + O(\| u\| ^\alpha) \quad (0 \leq \alpha < 1, \;0 \leq \mu < \frac\rho2) \] or even (the resonance case) \[ \left\| N(u) - \frac\rho2 Ju\right\| \leq \frac\rho2 \| Ju\| + O(\| u\| ^\alpha) \quad (0 \leq \alpha < 1) \] with some additional condition and describe those \(\rho\) for which \(\text{Im}\,(L - N) = H\) or \(\overline{\text{Im}\,(L - N)} = H\). These results are based on the degree theory for the corresponding classes of operators; the definition of this degree is given. In particular, the case of two-component systems is studied. As an application, the Hammerstein equation \[ Lu(x) = g(x,u(x)) + h(x) \] is considered.