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$S$-shaped connected component for a nonlinear Dirichlet problem involving mean curvature operator in one-dimension Minkowski space

Man Xu|Ruyun Ma
대한수학회 (The Korean Mathematical Society)|2018.11|대한수학회보 (Bulletin of the KMS)|제55권 제6호|pp. 1891-1908

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키워드

bifurcationpositive solutionsMinkowski space$S$-shaped connected componentmean curvature operator
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초록

In this paper, we investigate the existence of an $S$-shaped connected component in the set of positive solutions of the Dirichlet problem of the one-dimension Minkowski-curvature equation $$ \left\{ \aligned &\Big(\frac{u'}{\sqrt{1-u'^2}}\Big)'+\lambda a(x)f(u)=0,\ \ \ x\in(0,1),\\ &u(0)=u(1)=0,\\ \endaligned \right. $$ where $\lambda$ is a positive parameter, $f\in C[0,\infty)$, $a\in C[0,1]$. The proofs of main results are based upon the bifurcation techniques.

Abstract

In this paper, we investigate the existence of an $S$-shaped connected component in the set of positive solutions of the Dirichlet problem of the one-dimension Minkowski-curvature equation $$ \left\{ \aligned &\Big(\frac{u'}{\sqrt{1-u'^2}}\Big)'+\lambda a(x)f(u)=0,\ \ \ x\in(0,1),\\ &u(0)=u(1)=0,\\ \endaligned \right. $$ where $\lambda$ is a positive parameter, $f\in C[0,\infty)$, $a\in C[0,1]$. The proofs of main results are based upon the bifurcation techniques.

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