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Critical exponent and minimization problem in \(\mathbb{R}^N\). (English) Zbl 1052.35033

Summary: Let \(K(x)=\exp (\frac{| x|^2} {4})\), for \(x\in\mathbb{R}^N\), \(L^q(K)= \{u:\mathbb{R}^N \to\mathbb{R};\int_{\mathbb{R}^N}| u |^qK <\infty\}\) and \(H^1(K)=\{u\in L^2(K);\;|\nabla u|\in L^2(K)\}\). We are concerned with the following minimization problem \[ \inf\left\{\int_{\mathbb{R}^N} |\nabla u|^2K-\lambda \int_{\mathbb{R}^N} u^2K;u\in H^1(K),\int_{\mathbb{R}^N}| u+\varphi |^{q_c} K=1 \right\}, \] where \(q_c=\frac {2N}{N-2}\), \(N\geq 3\), \(\lambda\in \mathbb{R}\) and \(\varphi\in C(\mathbb{R}^N)\) is such that \(K\varphi\in L^\infty (\mathbb{R}^N)\). We show that for \(\varphi\neq 0\), the infimum is achieved under some condition on \(\lambda\).

MSC:

35B33 Critical exponents in context of PDEs
49J10 Existence theories for free problems in two or more independent variables
35J20 Variational methods for second-order elliptic equations
49J20 Existence theories for optimal control problems involving partial differential equations
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