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Zbl 0696.35042
Bensoussan, A.; Boccardo, L.; Murat, F.
On a nonlinear partial differential equation having natural growth terms and unbounded solution. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, No.4, 347-364 (1988). ISSN 0294-1449

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 $W\sb 0\sp{1,p}(\Omega)$ into $W\sp{-1,p'}(\Omega)$ and g is a nonlinear term with ``natural'' growth with respect to Du [i.e. such that $\vert g(x,u,\xi)\vert \le b(\vert u\vert)(\vert \xi \vert\sp p+c(x))]$, satisfying the sign condition g(x,u,$\xi)$u$\ge 0$ but no growth condition with respect to u. Here h belongs to $W\sp{-1,p'}(\Omega)$, thus the solution u of the problem does not in general be more smooth than $W\sb 0\sp{1,p}(\Omega)$. The existence of a solution is also proved for the corresponding obstacle problem.

MSC 2000:: *35J20 Second order elliptic equations, variational methods
35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J85 Unilateral problems; variational inequalities (elliptic type)
47J05 Equations involving nonlinear operators (general)
49J40 Variational methods including variational inequalities

Keywords: unbounded solutions

Cited in: Zbl 1077.35046 Zbl 0859.35031

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