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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)

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