Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms (Q928123)

The authors prove uniqueness of weak solutions in \(W_0^{1,p}(\Omega)\) to a class of problems whose prototype is \[ \begin{multlined} -\text{div}((1+| \nabla u| ^2)^{(p-2)/2}\nabla u) - \text{div}(c(x)(1+| u| ^2)^{(\tau +1)/2}) \\ + b(x)(1+| \nabla u| ^2)^{(\sigma+1)/2} = f~~\text{in}~~ \mathcal{D}'(\Omega), \end{multlined} \] where \(\Omega \subset \mathbb{R}^n\) is bounded, \(p>2N/(N+1)\), the coefficients \(c(x)\) and \(b(x)\) belong to suitable Lebesgue spaces, \(f\) is an element of the dual space \(W^{-1,p'}(\Omega)\) and \(\tau\) and \(\sigma\) are positive constants lying in some suitable intervals. The authors quote that the main difficulty in dealing with the above problem is the presence of the two lower terms \(b(x)(1+| \nabla u| ^2)\) and \(\text{div}(c(x)(1+| u| ^2)^{(\tau +1)/2})\) which in general produce a lack of coercivity. The uniqueness results of the paper are proved by two main steps. Firstly they prove a priori estimate of the ``reminder'' term \(S_m(u-v)\) of the difference of two solutions \(u\) and \(v\) of the problem. Then they derive a ``log-type estimate'' of the two solutions which provides the equality almost everywhere.