Multiplicity of positive solutions for semilinear elliptic problems with antipodal symmetry (Q2386875)

Let \(\Omega\) be a bounded smooth domain in \(\mathbb R^N\), \(N\geq 3\), and let \(f\in L^2(\Omega)\) with \(f\geq 0\). This paper concerns existence and multiplicity of positive solutions of the problem \[ \begin{cases} -\Delta u=| u| ^{2^*-2}u+f&\text{in }\Omega, \\ u=0&\text{on } \partial\Omega,\end{cases} \] where \(2^*=2N/(N-2)\). Results of existence and nonexistence of this problem are well known when \(f=0\). The author of the present paper investigates the case \(\Omega=-\Omega\) and there exists \(r>0\) such that \(B_r(0)\cap\Omega =\varnothing\). Also the function \(f\) is assumed to have antipodal symmetry, that is \(f(x)=f(-x)\). Classes of domains \(\Omega\) are described such that, if \(| f| _2\) is small then the problem above possesses multiple solutions.