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Supercritical elliptic problems involving a Cordes like operator. (English) Zbl 1466.35107

Summary: In this work we obtain positive bounded solutions of various perturbations of \[ \begin{cases} -\Delta u-\gamma\sum_{i,j=1}^N\frac{x_ix_j}{|x|^2}u_{x_i x_j} & = u^p\quad\text{ in }B_1,\\ u&=0\quad\text{on }\partial B_1, \end{cases}\tag{1} \] where \(B_1\) is the unit ball in \(\mathbb{R}^N\) where \(N\geq 3\), \(\gamma>0\) and \(1<p<p_{N,\gamma}\) where \[ p_{N, \gamma}:=\left\{\begin{array}{lc}\frac{N+2+3\gamma}{N-2-\gamma} & \text{ if }\gamma<N-2, \\ \infty & \text{ if }\gamma\geq N-2.\end{array}\right. \] Note for \(\gamma>0\) this allows for supercritical range of \(p\).

MSC:

35J15 Second-order elliptic equations
35J61 Semilinear elliptic equations
35B09 Positive solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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