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Keywords:
Quasi-linear elliptic problem; $(p,q)$-Laplacian operator; Critical Sobolev-Hardy exponent; Starshaped domain.
Quasi-linear elliptic problem; $(p,q)$-Laplacian operator; Critical Sobolev-Hardy exponent; Starshaped domain.
Summary:
Let $\Omega \subset \mathbb{R}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem \begin{align*} -\Delta_p u-\Delta_q u = \lambda ({\bf x} )\lvert u\rvert^{p^\star -2} u+\mu |u|^{r-2} u \end{align*} where $\mu$ is a positive parameter, $1 < q \le p < n$, $r\ge p^{\star}$ and $p^{\star}:=\frac{np}{n-p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p, q)$-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.
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