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Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. (English) Zbl 0965.35067
The authors consider the problem \[ -\Delta_p u=f_\lambda(u)\quad\text{in }\Omega,\quad u|_{\partial\Omega}= 0,\tag{1} \] where \(f_\lambda(u)=|u|^{r-2}u+ \lambda|u|^{q-2}u\) with \(1< q< p< r< p^*\), \(\lambda>0\) and \(\Omega\) a smooth bounded domain of \(\mathbb{R}^N\). They prove that there exists \(\Lambda>0\) such that:
1. If \(\lambda>\Lambda\) problem (1) has no positive solution \(u\in W^{1,p}_0(\Omega)\).
2. If \(0<\lambda<\Lambda\) problem (1) has at least two positive solutions \(u\in W^{1,p}_0(\Omega)\).
3. If \(\lambda= \Lambda\) there exists at least one positive solution to problem (1).
The main tool is an extension of a result by H. Brezis and L. Nirenberg [C. R. Acad. Sci. Paris, Sér. I 317, No. 5, 465-472 (1993; Zbl 0803.35029)].

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
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