POSITIVE SOLUTIONS FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

doi:10.1007/s10473-022-0507-z

Abstract

Abstract: This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms: $$ \left\{ \begin{array}{ll} -\Big(a+b \int_{\mathbb{R}^3} | \nabla u |^2 {\rm d}x\Big) \Delta u+ V(x)u=Q(x)u^p+\varepsilon f(x),\quad &x\in \mathbb{R}^3, \\ u>0,\quad &u\in H^1(\mathbb{R}^3). \end{array} \right.$$ Here $a,b$ are positive constants, $V(x),Q(x)$ are positive radial potentials, 1<p<5, $\varepsilon$ >0 is a small parameter, $f(x)$ is an external source term in $L^2(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$.

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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