New Developments on Nirenberg's Problem for Compact Perturbations of Quasimonotone Expansive Mappings in Reflexive Banach Spaces

Abstract

Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^*$. Let $T:X\to X^*$ be demicontinuous, quasimonotone and $\alpha$-expansive, and $C: X\to X^*$ be compact such that either (i) $\langle Tx+Cx, x\rangle \geq -d\|x\|$ for all $x\in X$ or (ii) $\langle Tx+Cx, x\rangle \geq-d\|x\|^2$ for all $x\in X$ and some suitable positive constants $\alpha$ and $d.$ New surjectivity results are given for the operator $T+C.$ The results are new even for $C=\{0\}$, which gives a partial positive answer for Nirenberg's problem for demicontinuous, quasimonotone and $\alpha$-expansive mapping. Existence result on the surjectivity of quasimonotone perturbations of multivalued maximal monotone operator is included. The theory is applied to prove existence of generalized solution in $H^{1}_{0}(\Omega)$ of nonlinear elliptic equation of the type \begin{align*} \begin{split} \left\{\begin{array}{cc} -\sum\limits_{i=1}^{N}{\frac{\partial}{\partial x_i} a_i(x, u(x), \nabla u(x))})+G_{\lambda}(x, u(x))=f(x) &\textrm{in $\Omega$}\\ u(x)=0&\textrm{$x\in\partial \Omega$},\\ \end{array}\right. \end{split} \end{align*} where $f\in L^{2}(\Omega)$, $\Omega$ is a nonempty, bounded and open subset of $\mathbb{R}^{N}$ with smooth boundary, $\lambda>0$, $ G_{\lambda}(x, u)=-div (\beta (\nabla u(x)))+\lambda u(x)+a_0(x, u(x), \nabla u(x))+g(x, u(x))$, $\beta: \mathbb{R}^{N}\to\mathbb{R}^{N}$, $a_i: \Omega\times \mathbb{R}\times \mathbb{R}^{N}\to\mathbb{R}$ ($i=0, 1, 2, ..., N$) and $g:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\to\mathbb{R}$ satisfy certain conditions.