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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
##### Keywords:
unbounded solutions
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##### References:
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