POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

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doi:10.1007/s10473-022-0507-z

Abstract: This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms:

{\begin{cases} - (a + b \int_{R^{3}} | \nabla u |^{2} d x) Δ u + V (x) u = Q (x) u^{p} + ε f (x), & x \in R^{3}, \\ u > 0, & u \in H^{1} (R^{3}) . \end{cases}

$\left\{ \begin{array}{ll} -\Big(a+b \int_{\mathbb{R}^3} | \nabla u |^2 {\rm d}x\Big) \Delta u+ V(x)u=Q(x)u^p+\varepsilon f(x),\quad &x\in \mathbb{R}^3, \\ u>0,\quad &u\in H^1(\mathbb{R}^3). \end{array} \right.$ Here

a, b

$a,b$ are positive constants,

V (x), Q (x)

$V(x),Q(x)$ are positive radial potentials, 1<p<5,

ε

$\varepsilon$ >0 is a small parameter,

f (x)

$f(x)$ is an external source term in

L^{2} (R^{3}) \cap L^{\infty} (R^{3})

$L^2(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$ .

Key words: nonlinear Kirchhoff problem, Lyapunov-Schmidt reduction, positive solutions, level set

CLC Number:

35J20

Narimane AISSAOUI, Wei LONG. POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS[J].Acta mathematica scientia,Series B, 2022, 42(5): 1817-1830.

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