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On a nonresonance condition for a semilinear elliptic problem. (English) Zbl 0735.35054
This paper is concerned with the semilinear elliptic problem \[ -\Delta u=f(u)+h(x) \hbox{ in }\Omega,\quad u=\varphi \hbox{ on }\partial\Omega, (1) \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^ N\) and \(f\) is a continuous function. The authors are interested in the conditions to be imposed on the nonlinearity \(f\) in order to guarantee nonresonance, i.e. the solvability of (1) for any \(h\) and \(\varphi\). A classical result asserts that if \(f\) satisfies a suitable polynomial growth restriction connected with the Sobolev embeddings and if \[ \limsup_{{s\to+\infty}\atop{s\to-\infty}}2F(s)s^{-2}<\lambda_ 1(\Omega) (2) \] then (1) is solvable for any \(h\in H^{- 1}(\Omega)\) and any \(\varphi\in H^{-1/2}(\partial\Omega)\), where \(F\) denotes the primitive \(F(s)=\int_ 0^ s f(t)dt\) and \(\lambda_ 1(\Omega)\) the first eigenvalue of \(-\Delta\) on \(H_ 0^ 1(\Omega)\).
The main result of this paper is: Suppose \[ \liminf_{{s\to+\infty}\atop{s\to-\infty}}2F(s)s^{-2}<\pi^ 2(2R(\Omega))^{-2},\quad p>N/2, (3) \] then problem (1) has a solution \(u\in W^{2,p}(\Omega)\) for any \(h\in L^ \infty(\Omega)\) and any \(\varphi\in W^{2-(1/p),p}(\partial\Omega)\), where \(R(\Omega)\) is the radius of the smallest open ball containing \(\Omega\).
For \(N>1\) we have \(\pi^ 2(2R(\Omega))^{-2}<\lambda_ 1(\Omega)\) and the question whether \(\pi^ 2(2R(\Omega))^{-2}\) can be replaced by \(\lambda_ 1(\Omega)\) in (3) remains open. No growth restriction connected with Sobolev imbeddings is imposed on \(f\).
Reviewer: M.A.Vivaldi (Roma)

35J65 Nonlinear boundary value problems for linear elliptic equations