Geodesic
$S$-shaped component of nodal solutions for problem involving one-dimension mean curvature operator
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 321-333
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Let $E=\{u\in C^1[0,1] \colon u(0)=u(1)=0\}$. Let $S_k^\nu $ with $\nu =\{+, -\}$ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^\nu $-solutions of the problem $$ \begin{cases} \biggl (\dfrac {u'}{\sqrt {1-u'^2}}\bigg )^{\prime }+\lambda a(x) f(u)=0, x\in (0,1), \\ u(0)=u(1)=0, \end{cases} $$ where $\lambda >0$ is a parameter, $a\in C([0, 1], (0,\infty ))$. We determine the intervals of parameter $\lambda $ in which the above problem has one, two or three $S_k^\nu $-solutions. The proofs of the main results are based upon the bifurcation technique.
Let $E=\{u\in C^1[0,1] \colon u(0)=u(1)=0\}$. Let $S_k^\nu $ with $\nu =\{+, -\}$ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^\nu $-solutions of the problem $$ \begin{cases} \biggl (\dfrac {u'}{\sqrt {1-u'^2}}\bigg )^{\prime }+\lambda a(x) f(u)=0, x\in (0,1), \\ u(0)=u(1)=0, \end{cases} $$ where $\lambda >0$ is a parameter, $a\in C([0, 1], (0,\infty ))$. We determine the intervals of parameter $\lambda $ in which the above problem has one, two or three $S_k^\nu $-solutions. The proofs of the main results are based upon the bifurcation technique.
DOI : 10.21136/CMJ.2023.0027-20
Classification : 34C10, 34C23, 35B40, 35J65
Keywords: mean curvature operator; $S_k^\nu $-solution; bifurcation; Sturm-type comparison theorem 
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     title = {$S$-shaped component of nodal solutions for problem involving one-dimension mean curvature operator},
     journal = {Czechoslovak Mathematical Journal},
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     year = {2023},
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Ma, Ruyun; He, Zhiqian; Su, Xiaoxiao. $S$-shaped component of nodal solutions for problem involving one-dimension mean curvature operator. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 321-333. doi: 10.21136/CMJ.2023.0027-20

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