Existence and nonexistence results for anisotropic quasilinear elliptic equations (Q1886259)

The paper concerns the existence and non-existence of solutions to the anisotropic quasilinear elliptic problem \(-\sum_{i=1}^n\partial_i(| \partial_i u| ^{m_i-2}\partial_i u)=\lambda u^{p-1}\) in \(\Omega\), \(u\geq0\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), where \(n\geq2\), \(\Omega\subset\mathbb R^n\) is a bounded domain with a sufficiently smooth boundary, \(m_i>1\) for \(i=1,\dots,n\), \(\lambda>0\) and \(p>1\). The natural space for the solutions is the anisotropic Sobolev space \(W^{1,m}_0(\Omega)\), \(m=(m_1,\dots,m_n)\), with the norm \(\| u\|_{1,m}=\sum_{i=1}^n\|\partial_i u\|_{m_i}\). Similar problems have been studied in many papers, the existence of solutions used to be investigated by variational methods. The authors for the first time try to use -- in analogy with the standard semilinear problem for \(-\Delta u=f(x,u)\) -- minimax methods for this type of problems. They are also interested in the question if the critical exponent \(q\) for the embedding \(W^{1,m}_0(\Omega)\) into \(L^q(\Omega)\) is also the borderline between existence and non-existence results for solutions of such equations. There are two different situations: when the anisotropy is ``concentrated'', the critical exponent \(m^\star\) is given by the identity \(n/m^\star=\sum_{i=1}^n(1/m_i)-1\) and satisfies \(m^\star\geq m_+=\max\{m_1,\dots,m_n\}\), and when it is ``spread out'', i.e. \(m^\star<m_+\), and the critical exponent is \(m_+\). The non-existence result is proved for the case of concentrated anisotropy and for domains with the property of \(\alpha\)-starshapedness, which is the starshapedness suitably modified according to the anisotropy \(m\).