Dual ground state solutions for the critical nonlinear Helmholtz equation
2019 ◽
Vol 150
(3)
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pp. 1155-1186
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Keyword(s):
AbstractUsing a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation $$-\Delta u-k^2u = Q(x) \vert u \vert ^{2^*-2}u,\quad u\in W^{2,2^*}({\open R}^{N})$$for N ⩾ 4, where 2* : = 2N/(N − 2). The coefficient $Q \in L^{\infty }({\open R}^{N}){\setminus }\{0\}$ is assumed to be nonnegative, asymptotically periodic and to satisfy a flatness condition at one of its maximum points. The solutions obtained are so-called dual ground states, that is, solutions arising from critical points of the dual functional with the property of having minimal energy among all nontrivial critical points. Moreover, we show that no dual ground state exists for N = 3.
2019 ◽
Vol 109
(2)
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pp. 193-216
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Keyword(s):
2014 ◽
Vol 58
(2)
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pp. 305-321
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2019 ◽
Vol 150
(4)
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pp. 1737-1768
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2015 ◽
Vol 39
(9)
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pp. 2193-2201
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Keyword(s):
2012 ◽
Vol 142
(4)
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pp. 867-895
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2021 ◽
2018 ◽
Vol 17
(3)
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pp. 1121-1145
2016 ◽
Vol 09
(04)
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pp. 1432-1439
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