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On a nonlinear partial differential equation having natural growth terms and unbounded solution. (English) Zbl 0696.35042
Summary: We prove the existence of a solution of the nonlinear elliptic equation: \(A(u)+g(x,u,Du)=h(x)\), where A is a Leray-Lions operator from \(W_ 0^{1,p}(\Omega)\) into \(W^{-1,p'}(\Omega)\) and g is a nonlinear term with “natural” growth with respect to Du [i.e. such that \(| g(x,u,\xi)| \leq b(| u|)(| \xi |^ p+c(x))]\), satisfying the sign condition g(x,u,\(\xi)\)u\(\geq 0\) but no growth condition with respect to u. Here h belongs to \(W^{-1,p'}(\Omega)\), thus the solution u of the problem does not in general be more smooth than \(W_ 0^{1,p}(\Omega)\). The existence of a solution is also proved for the corresponding obstacle problem.

MSC:
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
47J05 Equations involving nonlinear operators (general)
49J40 Variational inequalities
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