an:04212592
Zbl 0733.35043
Wang, Zhi Qiang
On a superlinear elliptic equation
EN
Ann. Inst. Henri Poincar??, Anal. Non Lin??aire 8, No. 1, 43-57 (1991).
00156520
1991
j
35J65 58E05
superlinear; subcritical growth; Dirichlet problem; three nontrivial solutions
The author deals with the following problem:
\[
-\Delta u=f(u)\text{ in } \Omega,\quad u=0\text{ on } \partial \Omega,
\]
where \(\Omega\) is a bounded domain in \(R^ n\) with regular boundary, assuming that
(f1) \(f\in C^ 1(R,R)\), \(f(0)=f'(0)=0;\)
(f2) There are constants \(C_ 1,C_ 2\) such that
\[
| f(t)| \leq C_ 1+C_ 2| t|^{\alpha},\quad 1<\alpha <(n+2)/(n-2)
\]
(f3) There are constants \(\mu >2\), \(M>0\) such that
\[
0<\mu F(t)\leq tf(t),\quad | t| \geq M,\text{ where } F(t)=\int^{t}_{0}f(r)dr.
\]
The main result is
Theorem. If f satisfies (f1)(f2)(f3), then the problem above possesses at least three nontrivial solutions.
In a classical paper, Ambrosetti and Rabinowitz obtained two nontrivial solutions, and infinitely many in the case of odd nonlinearities f. Infinitely many solutions can be obtained in case \(n=1\). The author establishes existence of multiple solutions in case \(n\geq 2\) without assuming any symmetry.
J.E.Bouillet (Buenos Aires)