Demicontinuity implies continuity
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I want to check the continuity of the operator $A:Xto X^{*}$ defined by $$langle A(u),vrangle=int_{Omega}a(x,nabla u)cdotnabla v,dx-int_{Omega}g_k(u)v,dxquadtext{ for every }u,vin X.$$ Here $Omega$ is a smooth bounded domain in $mathbb{R}^N$ with $Ngeq 2$ , $X=W_0^{1,p}(Omega)$ ( $p>1$ ) and $X^{*}$ is the dual space of $X$ . Moreover, $a(x,s)$ is a Caratheodory function defined on $Omegatimesmathbb{R}^N$ to $mathbb{R}^N$ , and $g_k(s)=text{min}{s^{-delta},k}$ for $k>0$ and $s>0$ and $g_k(s)=k$ for $sleq 0$ . One can take $A(x,zeta)=|zeta|^{p-2}zeta$ for example. To proceed, we need to prove that for every $u_nto u$ in $X$ , $$A(u_n)to A(u);text{ in }X^{*}.$$ I am able to prove that for every $u_nto u$ in $X$ , one has $$langle A(u_n),vrangleto langle A(u),vranglequadtext{...
