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Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in \(L^1(\Omega)\). (English) Zbl 1092.35032

Summary: We prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is \[ - \text{div}(a(x)(1+|\nabla u|^2)^{\frac {p-2}{2}}\nabla u) +b(x)(1+|\nabla u|^2)^{\frac {\lambda }{2}} =f \hbox { in }\Omega,\qquad u=0 \text{ on } \partial \Omega, \] where \(\Omega \) is a bounded open subset of \(\mathbb R^N\), \(N\geq 2\), \(2-1/N< p< N\), \(a\) belongs to \(L^{\infty }(\Omega )\), \(a(x) \geq \alpha_0>0\), \(f\) is a function in \(L^1(\Omega ) \), \(b\) is a function in \(L^r(\Omega )\) and \(0\leq \lambda <\lambda^*(N,p,r),\) for some \(r\) and \(\lambda^*(N,p,r)\).

MSC:

35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
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References:

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