On Nirenberg’s Problem for Compact Perturbations of Expansive Operator and an Application to p-Laplacian Type Equation with Nonmonotone Convection Function

Abstract

Let X be a real Banach space and $T: X\rightarrow X$ be a $\beta$-expansive and continuous operator. When X is a real Hilbert space, Nirenberg proposed a problem as to whether or not T is surjective provided that R(T) (range of T) has a nonempty interior. Recently, Asfaw gave a positive solution for this problem provided that $\overset{\circ }{R(T)}$ is unbounded. In this paper, a new surjectivity result for T is proved provided that X is a real Banach space and the condition on $\overset{\circ }{R(T)}$ is replaced by the condition that there exists a suitable nonnegative constant $\alpha$ satisfying $\langle Tx-Ty, j(x-y)\rangle \ge -\alpha \Vert x-y\Vert ^2$ for all $(x, y)\in X^{2}$ and some $j(x-y)\in J(x-y)$, where J is the duality mapping on X. Existence theorems are derived for compact perturbations of T. The results are applied to establish existence of solution (weak) in $W^{1,p}_{0}(\Omega )$ (with $p\ge 2$) for the nonlinear equation given by

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta _{p}a(u(x)) +\varepsilon F(x,u(x), \nabla u(x))=0,~~x\in \Omega ,\\ u(x)=0,~~x\in \partial \Omega ,\\ \end{array}\right. \end{aligned}$$

where $\Omega$ is a nonempty, bounded and open subset of ${\textbf{R}}^{N}$ (for $N\ge 1$) with smooth boundary, the convection term F is Lipschitz continuous (not necessarily monotone), $a:{\textbf{R}}\rightarrow {\textbf{R}}$ satisfies suitable conditions and $\varepsilon >0$ is sufficiently small.