Kolonitskii, S. B. Multiplicity of 1D-concentrated positive solutions to the Dirichlet problem for an equation with \(p\)-Laplacian. (English. Russian original) Zbl 1326.35162 Funct. Anal. Appl. 49, No. 2, 151-154 (2015); translation from Funkts. Anal. Prilozh. 49, No. 2, 88-92 (2015). Summary: 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 \(1 < p < q < p^\ast_{n-1}\), where \(p^\ast_{n-1}\) is the critical Sobolev exponent for embedding in \(\mathbb{R}^{n-1}\) and either \(n = 4\) or \(n \geqslant 6\). 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. Cited in 2 Documents MSC: 35J92 Quasilinear elliptic equations with \(p\)-Laplacian 35B09 Positive solutions to PDEs Keywords:\(p\)-Laplacian; multiplicity of solutions PDFBibTeX XMLCite \textit{S. B. Kolonitskii}, Funct. Anal. Appl. 49, No. 2, 151--154 (2015; Zbl 1326.35162); translation from Funkts. Anal. Prilozh. 49, No. 2, 88--92 (2015) Full Text: DOI arXiv References: [1] Byeon, J., No article title, J. Differential Equations, 136, 136-165 (1997) · Zbl 0878.35043 [2] Coffman, C. V., No article title, J. Differential Equations, 54, 429-437 (1984) · Zbl 0569.35033 [3] Kolonitskii, S. B., No article title, Algebra i Analiz, 22, 206-221 (2010) [4] Li, Y. Y., No article title, J. Differential Equations, 83, 348-367 (1990) · Zbl 0748.35013 [5] Malchiodi, A., No article title, Boll. Unione Mat. Ital., 8, 615-628 (2005) · Zbl 1182.35121 [6] Mizoguchi, N.; Suzuki, T., No article title, Houston J. Math., 22, 199-215 (1996) · Zbl 0862.35036 [7] Nazarov, A. I., No article title, Proc. St.-Petersburg Math. Soc., 10, 33-62 (2004) [8] Nazarov, A. I., No article title, Probl. Math. Anal., 20, 171-190 (2000) · Zbl 1002.46025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.