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On quasilinear Brezis-Nirenberg type problems with weights. (English) Zbl 1209.35055

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 critical dimensions, similarly as in the celebrated result of H. Brézis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437–477 (1983; Zbl 0541.35029)].

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B09 Positive solutions to PDEs
35B07 Axially symmetric solutions to PDEs
35B33 Critical exponents in context of PDEs

Citations:

Zbl 0541.35029
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