Positive solutions for the nonhomogeneous \(p\)-Laplacian equation in \(\mathbb R^N\) (Q2409586)

The authors establish the following result: Let \(N\geq 2\) be an integer, and let \(p\in (1,N)\), \(q\in (1,p)\), and \(m\in [p,p^*)\), where \(p^*=\frac{pN}{N-p}\). Moreover, let \(f\in L^{\frac{q}{q-1}}(\mathbb{R}^N)\cap L^{\frac{p^*}{p^*-1}}(\mathbb{R}^N)\setminus\{0\}\) with \(f\geq 0\) in \(\mathbb{R}^N\). Then, the problem \[ \operatorname{div}(|\nabla u|^{p-2}\nabla u)+|u|^{m-2}u=|u|^{q-2}u+f(x)\text{ in }\mathbb{R}^N, \] \[ u>0\text{ in }\mathbb{R}^N,\quad u\in \big\{v\in L^q(\mathbb{R}^N)\cap L^{p^*}(\mathbb{R}^N): |\nabla u|\in L^p(\mathbb{R}^N)\big\} \] admits at least one solution \(u\) such that \[ \displaystyle{\int_{\mathbb{R}^N}(|\nabla u|^p+|u|^q)dx\rightarrow 0},\quad\text{as }\displaystyle{\int_{\mathbb{R}^N}|f|^{\frac{q}{q-1}}dx\rightarrow 0}.\tag{1} \] The existence of solutions is proved via minimization on the Nehari manifold and the strong maximum principle. The limit (1) follows from upper estimates of \(\int_{\mathbb{R}^N}|\nabla u|^pdx\) and \(\int_{\mathbb{R}^N}|u|^qdx\) in terms of \(\int_{\mathbb{R}^N}|f|^{\frac{q}{q-1}}dx\).