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Non-radial maximizers for functionals with exponential nonlinearity in \(\mathbb R ^2\). (English) Zbl 1096.35041

Let \(B=B(0,1)\) be the unit ball in \(\mathbb{R}^2\), and let \(F:H^1_0(B)\to \mathbb{R}\) be the functional \[ F(u)=\int_B| x| ^\alpha\Bigl(e^{p| u| ^\gamma}-1-p| u| ^\gamma\Bigr)\,dx, \] where \(\alpha>0,\) \(p>0\) and \(1<\gamma\leq 2\). The interest of the authors is to understand for which values of \(\alpha\), \(p\) and \(\gamma\) the supremum of \(F(u)\) on the set \(S_1=\{u\in H^1_0:\| u\|_{H^1_0}\leq 1\}\) is finite and attained. A second question is to know whether such a supremum is achieved by a radial or by a nonradial function. Let \[ T_{\alpha,p,\gamma}=\sup_{u\in S_1}F(u),\;\;\;T^R_{\alpha,p,\gamma}= \sup_{u\in S_1, u \;rad}F(u), \] where \(u\;rad\) means \(u\) radially symmetric. The main results are the following. For \(p>0\), \(1<\gamma<2\), and for \(0<p\leq 4\pi\), \(\gamma=2\), there exists \(\alpha^*>0\) such that \(T_{\alpha,p,\gamma}>T^R_{\alpha,p,\gamma}\) for any \(\alpha>\alpha^*\). Moreover, for \(p>0\), \(1<\gamma<2\), and for \(0<p< 4\pi\), \(\gamma=2\), there exists \(\alpha^*>0\) such that, for \(\alpha>\alpha^*\) no maximizer of \(F(u)\) in \(S_1\) is radial (symmetry breaking phenomena). Also the supercritical case \(\gamma=2\) and \(p>4\pi\) is discussed. Furthermore, the authors consider the functional \[ G_\lambda(u)=\int_B| x| ^\alpha\Bigl(e^{p| u| ^2}-1-\lambda p| u| ^2\Bigr)\,dx, \] where \(\lambda\in [0,1]\), \(\alpha>0\) and \(0<p\leq 4\pi\). If \[ T_{\alpha,\lambda}=\sup_{u\in S_1}G_\lambda(u),\;\;\;T^R_{\alpha,\lambda}=\sup_{u\in S_1,\;u\;rad}G_\lambda(u), \] the following results are proved: Let \(0<p\leq 4\pi\). Then there exists \(0\leq \lambda^*<1\) such that for any \(\lambda\in [\lambda^*,1]\), \(T_{\alpha,\lambda}>T^R_{\alpha,\lambda}.\) Furthermore, for \(0<p<4\pi\), there exists \(0\leq \lambda^*<1\) such that for any \(\lambda\in [\lambda^*,1]\), no maximizer of \(G_\lambda(u)\) on \(S_1\) is radial for \(\alpha\) large.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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