Remark on the structure of the range of second order nonlinear elliptic operator. (English) Zbl 0702.35090

Drábek, Pavel; Tomiczek, Petr

Commentat. Math. Univ. Carol. 30, No. 3, 455-464 (1989).

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

Keywords:

semilinear problems; nonlinearities with linear growth; strong solution; first eigenvalue