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Remark on the structure of the range of second order nonlinear elliptic operator. (English) Zbl 0702.35090
The paper is concerned with the boundary value problem $(*)\quad Lu+\lambda_ 1u+g(x,u)=s\phi +h\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega$ where $$L=\partial_ ia_{ik}(x)\partial_ k-a_ 0(x)$$ is a uniformly elliptic operator with $$C^ 1$$-smooth $$a_{ik}$$ and $$a_ 0$$ in $$L^{\infty}(\Omega)$$, $$\Omega \subset {\mathbb{R}}^ N$$ a bounded domain. $$\lambda_ 1$$ is the first eigenvalue of -L with eigenfunction $$\phi$$. The solvability of (*) in $$W^{2,p}(\Omega)\cap W_ 0^{1,2}(\Omega),$$ $$p>N$$, is analyzed for $$h\in L^ p(\Omega)\cap$$ range of $$-L+\lambda_ 1$$ and $$s\in {\mathbb{R}}$$ varying in a certain interval, if the Carathéodory function g satisfies:
(i) $$\lim_{u\to \pm \infty} g(,u)/u\in L^ p(\Omega)$$, (ii) g(x,u)u$$\geq 0$$, (iii) further sign conditions.
Thus results from R. Iannacci, M. N. Nkashama and J. R. Ward jun., Trans. Am. Math. Soc. 311, No.2, 711-726 (1989; Zbl 0686.35045)] are completed.
Reviewer: R.Racke

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations
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