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# Gagliardo-Nirenberg-Sobolev inequalities on planar graphs

• In this paper we study a family of the Gagliardo-Nirenberg-Sobolev interpolation inequalities on planar graphs. We are interested in knowing when the best constants in the inequalities are achieved. The inequalities being equivalent to some minimization problems, we also analyse the set of solutions of the Euler-Lagrange equations satisfied by extremal functions, or equivalently, by minimizers.

Mathematics Subject Classification: Primary: 49J40, 35R45, 35A23, 34B45; Secondary: 34K38, 39B05, 37E25.

 Citation: • • Figure 1.  Phase plane for the dynamical system $v'=u,\; -u'+v= |v|^{p-2}v\;\; \mbox{in}\;\; {\mathbb R}$, with $p=3$

Figure 2.  Solution of the Kirchhoff-ODE system for $p=3$, and edges of lengths $1, 5$ and $+\infty$. The dotted curves correspond to the functions $v_1$, the dot-dashed to $v_2$ and the full line to $v_0$. The dots in the graphics correspond in the phase plane to the values of $v_0, v_1$ and $v_2$ at the origin: the abscissas of the three dots coincide and the sum of the outward derivatives, corresponding to $-$ the sum of the ordinates of the dots, is equal to $0$

Figure 3.  Solution of the Kirchhoff-ODE system for $p=3$, and edges of lengths $1, 2$ and $5$. The dotted curve corresponds to $v_1$ ($\ell_1=1$), the dot-dashed one to $v_3$ ($\ell_3=5$) and the full line to $v_2$ ($\ell_2=2$). We observe that this solution is near the solution to the $1, 5, +\infty$ problem on the right of Figure 2. On the contrary, it is difficult to imagine a solution near the one on the left of Figure 2 for $\ell_2=2$. Indeed, if $\ell_2$ were larger, we would be able to find one such solution, making the number of solutions larger than or equal to $2$. We see this in the next figure, where $\ell_2= 4$

Figure 4.  Solution of the Kirchhoff-ODE system for $p=3$, and edges of lengths $1, 4$ and $5$. The dotted curves correspond to the functions $v_1$ ($\ell_1=1$), the dot-dashed to $v_3$ ($\ell_3=5$) and the full line to $v_2$ ($\ell_2=4$). The dots in the graphics correspond to the origin : the values of $v_1(0), v_2(0)$ and $v_3(0)$ (the abscissas of the three dots) coincide and the sum of the outward derivatives, corresponding to $-$ the sum of the ordinates of the dots, is equal to $0$. Note that the figure on the left has the same structure as the left figure on Figure 2, and the same analogy is found for the figures on the right

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