## 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

Zbl 0541.35029