Geodesic
Multiplicity of 1D-concentrated positive solutions to the Dirichlet problem for an equation with $p$-Laplacian
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 88-92 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the Dirichlet problem for the equation $-\Delta_p = u^{q-1}$ with $p$-Laplacian in a thin spherical annulus in $\mathbb R^n$ with $1 p q p^*_{n-1}$, where $p^*_{n-1}$ is the critical Sobolev exponent for embedding in $\mathbb R^{n-1}$ and either $n=4$ or $n \ge 6$. We prove that this problem has a countable set of solutions concentrated in neighborhoods of certain curves. Any two such solutions are nonequivalent if the annulus is thin enough. As a corollary, we prove that the considered problem has as many solutions as required, provided that the annulus is thin enough.
Keywords: $p$-Laplacian, multiplicity of solutions. 
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     author = {S. B. Kolonitskii},
     title = {Multiplicity of {1D-concentrated} positive solutions to the {Dirichlet} problem for an equation with $p${-Laplacian}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {88--92},
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S. B. Kolonitskii. Multiplicity of 1D-concentrated positive solutions to the Dirichlet problem for an equation with $p$-Laplacian. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 88-92. http://geodesic.mathdoc.fr/item/FAA_2015_49_2_a10/

[1] J. Byeon, J. Differential Equations, 136:1 (1997), 136–165 | DOI | MR | Zbl

[2] C. V. Coffman, J. Differential Equations, 54 (1984), 429–437 | DOI | MR | Zbl

[3] S. B. Kolonitskii, Algebra i analiz, 22:3 (2010), 206–221 | MR

[4] Y. Li, J. Differential Equations, 83:2 (1990), 348–367 | DOI | MR | Zbl

[5] A. Malchiodi, Boll. Unione Mat. Ital., Ser. B, 8:3 (2005), 615–628 | MR | Zbl

[6] N. Mizoguchi, T. Suzuki, Houston J. Math., 22:1 (1996), 199–215 | MR | Zbl

[7] A. I. Nazarov, Trudy S.-Peterburg. MO, 10 (2004), 33–62

[8] A. I. Nazarov, Probl. matem. analiza, 20 (2000), 171–190 | Zbl