Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

Abstract

Let $X$ be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space $X^{\ast }$. Let $T:X\supseteq D(T)\to 2^{X^{\ast }}$ be maximal monotone and $S:X\supseteq D(S)\to X^{\ast }$ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space $W\subset D(S)$, dense and continuously embedded in $X$. Assume, further, that there exists $d\ge 0$ such that $\langle v^{\ast }+Sx,x\rangle \ge -$d${‖x‖}^2$ for all $x\in D(T)\cap D(S)$ and $v^{\ast }\in Tx$. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type $T+S$. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator $L:X\supseteq D(L)\to X^{\ast }$ is given as a result of surjectivity of $L+S$, where $S$ is of type $(M)$ with respect to $L$. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in $X=L^p(0,T;W_0^{1,p}(\Omega ))$ of a nonlinear parabolic problem of the type $u_t-{\sum }_{i=1}^n(\partial /\partial x_i)a_i(x,t,u,\nabla u)=f(x,t)$, $(x,t)\in Q$; $u(x,t)=0$, $(x,t)\in \partial \Omega \times (0,T)$; $u(x,0)=0$, $x\in \Omega$, where $p>1$, $\Omega$ is a nonempty, bounded, and open subset of ${\mathbb{R}}^N$, $a_i:\Omega \times (0,T)\times \mathbb{R}\times {\mathbb{R}}^N\to \mathbb{R}$ $(i=\mathrm{1,2},\dots ,n)$ satisfies certain growth conditions, and $f\in L^{p^{\prime }}(Q)$, $Q=\Omega \times (0,T)$, and $p^{\prime }$ is the conjugate exponent of $p$.