Electronic Journal of Differential Equations (Jun 2017)
Nontrivial solutions of inclusions involving perturbed maximal monotone operators
Abstract
Let X be a real reflexive Banach space and $X^*$ its dual space. Let $L: X\supset D(L)\to X^*$ be a densely defined linear maximal monotone operator, and $T:X\supset D(T)\to 2^{X^*}$, $0\in D(T)$ and $0\in T(0)$, be strongly quasibounded maximal monotone and positively homogeneous of degree 1. Also, let $C:X\supset D(C)\to X^*$ be bounded, demicontinuous and of type $(S_+)$ w.r.t. to D(L). The existence of nonzero solutions of $Lx+Tx+Cx\ni0$ is established in the set $G_1\setminus G_2$, where $G_2\subset G_1$ with $\overline G_2\subset G_1$, $G_1, G_2$ are open sets in X, $0\in G_2$, and $G_1$ is bounded. In the special case when L=0, a mapping $G:\overline G_1\to X^*$ of class (P) introduced by Hu and Papageorgiou is also incorporated and the existence of nonzero solutions of $Tx+ Cx+ Gx\ni 0$, where T is only maximal monotone and positively homogeneous of degree $\alpha\in (0, 1]$, is obtained. Applications to elliptic partial differential equations involving p-Laplacian with $p \in (1, 2]$ and time-dependent parabolic partial differential equations on cylindrical domains are presented.
