Existence and asymptotic behavior of high energy normalized solutions for the Kirchhoff type equations in

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In this paper, we study the multiplicity of solutions with a prescribed 054989" title="Click to view the MathML source">L²

L^{2}

-norm for a class of nonlinear Kirchhoff type problems in R³

R^{3}

05">

−(a+b∫_R³|∇u|²)Δu−λu=|u|^p−2u,

- (a + b \int_{R^{3}} {| \nabla u |}^{2}) Δ u - λ u = {| u |}^{p - 2} u,

where a,b>0

a, b > 0

are constants, λ∈R

λ \in R

p \in (\frac{14}{3}, 6)

. To get such solutions we look for critical points of the energy functional

I_{b} (u) = \frac{a}{2} \int_{R^{3}} {| \nabla u |}^{2} + \frac{b}{4} {(\int_{R^{3}} {| \nabla u |}^{2})}^{2} - \frac{1}{p} \int_{R^{3}} {| u |}^{p}

restricted on the following set

S_{r} (c) = {u \in H_{r}^{1} (R^{3}) : {‖ u ‖}_{L^{2} (R^{3})}^{2} = c}, c > 0 .

For the value 05eb3b">

p \in (\frac{14}{3}, 6)

considered, the functional I_b

I_{b}

is unbounded from below on S_r(c)

S_{r} (c)

. By using a minimax procedure, we prove that for any e15e82f42b5e" title="Click to view the MathML source">c>0

c > 0

, there are infinitely many critical points 5e837cb62a1196">

{u_{n}^{b}}_{n \in N^{+}}

of I_b

I_{b}

restricted on S_r(c)

S_{r} (c)

with the energy

I_{b} (u_{n}^{b}) \to + \infty (n \to + \infty)

. Moreover, we regard b

b

as a parameter and give a convergence property of

u_{n}^{b}

as 052e06" title="Click to view the MathML source">b→0⁺

b \to 0^{+}

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