On the Helmholtz equation and Dancer’s-type entire solutions for nonlinear elliptic equations

Liu, Yong; Wei, Juncheng

doi:10.1090/proc/14278

Cited by 7 publications

(2 citation statements)

References 20 publications

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“…On the other hand, Evequoz and Weth [5] used mountain pass techniques to find nonperturbative solutions far from the zero solution. These approaches have been extended in various ways in [22,21]. In [4] the topology of the zero level sets of bounded real solutions to (\Delta -1)u + u 3 = 0 are studied.…”

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confidence: 99%

Existence and Asymptotics of Nonlinear Helmholtz Eigenfunctions

Gell-Redman¹,

Hassell²,

Shapiro³

et al. 2020

SIAM J. Math. Anal.

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We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form (\Delta -\lambda 2 )u = N [u], where \Delta = -\sum j \partial 2 j is the Laplacian on \BbbR n , \lambda is a positive real number, and N [u] is a nonlinear operator depending polynomially on u and its derivatives of order up to order two. Nonlinear Helmholtz eigenfunctions with N [u] = \pm | u| p - 1 u were first considered by Guti\' errez [Math. Ann., 328 (2004), pp. 1--25]. We show that for suitable nonlinearities and for every f \in H k+4 (\BbbS n - 1 ) of sufficiently small norm, there is a nonlinear Helmholtz function taking the form u(r, \omega ) = r - (n - 1)/2 (e - i\lambda r f (\omega ) + e +i\lambda r b(\omega ) + O(r - \epsilon )), as r \rightar \infty , \epsilon > 0, for some b \in H k (\BbbS n - 1 ). Moreover, we prove the result in the general setting of asymptotically conic manifolds. The proof uses an elaboration of anisotropic Sobolev spaces defined by Vasy [A minicourse on microlocal analysis for wave propagation, in Asymptotic Analysis in General Relativity, London Math. Soc. Lecture Note Ser. 443, Cambridge University Press, Cambridge, 2018, pp. 219--374], between which the Helmholtz operator \Delta -\lambda 2 acts invertibly. These spaces have a variable spatial weight \sansl \pm , varying in phase space and distinguishing between the two ``radial sets"" corresponding to incoming oscillations, e - i\lambda r , and outgoing oscillations, e +i\lambda r . Our spaces have, in addition, module regularity with respect to two different ``test modules"" and have algebra (or pointwise multiplication) properties that allow us to treat nonlinearities N [u] of the form specified above.

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mentioning

confidence: 99%

Existence and Asymptotics of Nonlinear Helmholtz Eigenfunctions

Gell-Redman¹,

Hassell²,

Shapiro³

et al. 2020

SIAM J. Math. Anal.

View full text Add to dashboard Cite

show abstract

“…Nonperturbative were found by Evequoz and Weth [6], who used mountain pass techniques to find solutions far from the zero solution. These approaches have been extended in various ways in [21,20]. In [5] the topology of the zero level sets of bounded real solutions to (∆ − 1)u + u 3 = 0 are studied.…”

Section: Introductionmentioning

confidence: 99%

Existence and asymptotics of nonlinear Helmholtz eigenfunctions

Gell-Redman¹,

Hassell²,

Shapiro³

et al. 2019

Preprint

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We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the formwhere ∆ = − j ∂ 2 j is the Laplacian on R n with sign convention that it is positive as an operator, λ is a positive real number, and N [u] is a nonlinear operator that is a sum of monomials of degree ≥ p in u, u and their derivatives of order up to two, for some p ≥ 2. Nonlinear Helmholtz eigenfunctions with N [u] = ±|u| p−1 u were first considered by Gutiérrez [10]. Such equations are of interest in part because, for certain nonlinearities N [u], they furnish standing waves for nonlinear evolution equations, that is, solutions that are time-harmonic.We show that, under the condition (p − 1)(n − 1)/2 > 2 and k > (n − 1)/2, for every f ∈ H k+2 (S n−1 ) of sufficiently small norm, there is a nonlinear Helmholtz function taking the form u(r, ω) = r −(n−1)/2 e −iλr f (ω) + e +iλr g(ω) + O(r − ) , as r → ∞, > 0, for some g ∈ H k (S n−1 ). Moreover, we prove the result in the general setting of asymptotically conic manifolds. The proof uses an elaboration of anisotropic Sobolev spaces X s,l ± , Y s,l ± , defined by Vasy [32], between which the Helmholtz operator ∆ − λ 2 acts invertibly. These spaces have a variable spatial weight l ± , varying in phase space and distinguising between the two 'radial sets' corresponding to incoming oscillations, e −iλr , and outgoing oscillations e +iλr . Our spaces have, in addition, module regularity with respect to two different 'test modules', and have algebra (or pointwise multiplication) properties which allow us to treat nonlinearities N [u] of the form specified above.

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The nonlinear Schrödinger equation in the half-space

Fernández

Weth

2021

Math. Ann.

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The present paper is concerned with the half-space Dirichlet problem "Equation missing"where $$\mathbb {R}^{N}_{+}:= \{\,x \in \mathbb {R}^N: x_N > 0\, \}$$ R + N : = { x ∈ R N : x N > 0 } for some $$N \ge 1$$ N ≥ 1 and $$p > 1$$ p > 1 , $$c > 0$$ c > 0 are constants. We analyse the existence, non-existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ). We prove that the existence and multiplicity of bounded positive solutions to ($$P_c$$ P c ) depend in a striking way on the value of $$c > 0$$ c > 0 and also on the dimension N. We find an explicit number $${c_p}\in (1,\sqrt{e})$$ c p ∈ ( 1 , e ) , depending only on p, which determines the threshold between existence and non-existence. In particular, in dimensions $$N \ge 2$$ N ≥ 2 , we prove that, for $$0< c < {c_p}$$ 0 < c < c p , problem ($$P_c$$ P c ) admits infinitely many bounded positive solutions, whereas, for $$c > {c_p}$$ c > c p , there are no bounded positive solutions to ($$P_c$$ P c ).

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On the Helmholtz equation and Dancer’s-type entire solutions for nonlinear elliptic equations

Abstract: Starting from a bound state (positive or sign-changing) solution toand solutions to the Helmholtz equation ∆u 0 + λu 0 = 0 in R n , λ > 0, we build new Dancer's type entire solutions to the nonlinear scalar equation −∆u = |u| p−1 u − u in R m+n .

Cited by 7 publications

References 20 publications

Existence and Asymptotics of Nonlinear Helmholtz Eigenfunctions

Existence and Asymptotics of Nonlinear Helmholtz Eigenfunctions

Existence and asymptotics of nonlinear Helmholtz eigenfunctions

The nonlinear Schrödinger equation in the half-space

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