Compact embeddings and indefinite semilinear elliptic problems
The author is interested in finding weak nonnegative solutions of Emden-Fowler type problems \[ \begin{cases} -\Delta u-h(x)u^{p-1} =0\quad \text{in }\mathbb{R}^N\\ 0\lneqq u\in E:=D^{1,2} (\mathbb{R}^N)\cap L^p\bigl( \mathbb{R}^N,|h |\bigr), \end{cases}\tag{1} \] where \(N\geq 3\), \(p>2\) and \(h\in L^1_{\text{loc}} :h^+(x): =\max(0,h(x)) \not\equiv 0\). Here by \(D^{1,2}(\mathbb{R}^N)\) is denoted the closure \(C^\infty_{\text{c}} (\mathbb{R}^N)\) with respect to the norm \((\int|\nabla u |^2 dx)^{1/2}\) in \(L^{2^*}\). Furthermore, the author assumes \(h\) to be symmetric with respect to some compact subgroups \({\mathcal G}\) of \(O(N)\), the group of orthogonal linear transformations in \(\mathbb{R}^N\), that is \[ (g_*h)(x):= h \bigl(g^{-1}(x) \bigr)=h(x), \quad g\in{\mathcal G}\text{ a.e. in } \mathbb{R}^N. \]
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