On quasilinear Brezis-Nirenberg type problems with weights
The authors consider radially symmetric positive solutions of the \(p\)-Laplacian problem \[ \begin{cases} -\Delta_p u =\lambda C(| x|)| u|^{p-2}u +B(| x|)| u|^{q-2}u, &\text{in } B_R(0)\setminus\{0\},\\ u=0&\text{on } \partial B_R(0),\\ \lim_{| x|\to0}| x|^{N-1}|\nabla u(x)|^{p-1}=0, \end{cases}\tag{1} \] where \(B_R(0)\) denotes, for \(R>0\), the open ball in \(\mathbb{R}^N\) of radius \(R\), centered at \(0\). It is assumed that \(q\geq p>1\), \(N\geq p\), and that the weights \(B,C\) are positive and such that \(r^{N-1}B(r)\) and \(r^{N-1}C(r)\) are in \(L^1(0,R)\). First a critical exponent \(p^*\) is defined that depends on \(B\) and \(p\) and controls the compactness of embeddings of certain weighted Sobolev spaces. The existence of a smallest eigenvalue \(\lambda_1\), which is positive, and a corresponding eigenfunction \(\varphi_1\) for the nonlinear eigenvalue problem \[ \begin{cases} -\Delta_p u =\lambda C(| x|)| u|^{p-2}u &\text{in } B_R(0)\setminus\{0\},\\ u=0&\text{on } \partial B_R(0),\\ \lim_{| x|\to0}| x|^{N-1}|\nabla u(x)|^{p-1}=0, \end{cases}\tag{2} \] is proved under appropriate conditions on \(p\) and \(C\). With these notions, and imposing additional assumptions on the weights \(B\) and \(C\) and the exponents \(p\) and \(q\), the authors prove existence and nonexistence results roughly of the following form: The subcritical problem (1), where \(q<p^*\), has a nontrivial, radially symmetric, and nonnegative solution if and only if \(\lambda<\lambda_1\). In the critical case \(q=p^*\), if \(p\) and \(q\) satisfy a condition that restricts the dimension \(N\), there exist \(0<\lambda^*< \lambda^{**}<\lambda_1\) such that (1) has a nontrivial, radially symmetric, and nonnegative solution if \(\lambda\in(\lambda^{**},\lambda_1)\), and it has no such solution if \(\lambda\in(0,\lambda^*)\). Finally, (1) has no such solution if \(q\) is supercritical, i.e. \(q>p^*\), and \(R\) and \(\lambda>0\) are small enough. The dimension restriction in the critical case exhibits certain \textit{critical dimensions}, similarly as in the celebrated result of \textit{H. Brézis} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 36, 437--477 (1983; Zbl 0541.35029)].
scientific article; zbMATH DE number 5734996