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On existence and nonexistence of the positive solutions of non-newtonian filtration equation
The maximal number of interior peak solutions concentrating on hyperplanes for a singularly perturbed Neumann problem
1. | Institute of Mathematics, Hangzhou Dianzi Universitye, Xiasha Hangzhou Zhejiang 310018, China |
$\varepsilon^2 \Delta u-u+f(u)=0, u>0 $ in $B_1$,
$\frac{\partial u}{\partial \nu}=0 $ on $\partial B_1,$
where $\Delta = \sum_{i=1}^N \frac{\partial^2}{\partial x_i^2}$ is the Laplace operator, $B_1$ is the unit ball centered at the origin in $R^N$ $(N\ge 3)$, $\nu$ denotes the unit outer normal to $\partial B_1$, $\varepsilon > 0$ is a constant, and $f$ is a superlinear, subcritical nonlinearity . We will show that when $\e$ is sufficiently small there exists a solution with K interior peaks located on a hyperplane, where $1\le K \varepsilon\frac{C}{(\varepsilon)^{N-1}}$ with $C$ a positive constant depending on $N$ and $f$ only. As a consequence, we obtain that there exists at least $[\frac{C}{(\varepsilon)^{N-1}}]$ number of solutions for $\varepsilon$ sufficiently small.
References:
[1] |
A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Comm. Math. Phys, 235 (2003), 427-466.
doi: doi:10.1007/s00220-003-0811-y. |
[2] |
A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part II, Indiana Univ. Math. J, 53 (2004), 297-329.
doi: doi:10.1512/iumj.2004.53.2400. |
[3] |
W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, preprint 2010. |
[4] |
P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Diff. Eqns, 4 (1999), 1-69. |
[5] |
P. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eqns, 160 (2000), 283-356.
doi: doi:10.1006/jdeq.1999.3660. |
[6] |
M. del Pino and D. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898.
doi: doi:10.1512/iumj.1999.48.1596. |
[7] |
M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79.
doi: doi:10.1137/S0036141098332834. |
[8] |
M. del Pino, P. Felmer and J. Wei, On the role of distance function in some singularly perturbed problems, Comm. P.D.E., 25 (2000), 155-177.
doi: doi:10.1080/03605300008821511. |
[9] |
M. del Pino, P. Felmer and J. Wei, Multiple-peak solutions for some singular perturbation problems, Cal. Var. P.D.E., 10 (2000), 119-134.
doi: doi:10.1007/s005260050147. |
[10] |
E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific. J. Math., 189 (1999), 241-262.
doi: doi:10.2140/pjm.1999.189.241. |
[11] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^N$, in "Mathematical Analysis and Applications," Part A, Adv. Math. Suppl. Studies, Vol. 7A, pp.369-402, Academic Press, New York, 1981. |
[12] |
R. Gardner and L. A. Peletier, The set of positive solutions of semilinear equations in large balls, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 53-72. |
[13] |
C. Gui and J. Wei, Multiple interior spike solutions for some singularly perturbed Neumann problems, J. Diff. Eqns., 158 (1999), 1-27.
doi: doi:10.1016/S0022-0396(99)80016-3. |
[14] |
C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Can. J. Math., 52 (2000), 522-538.
doi: doi:10.4153/CJM-2000-024-x. |
[15] |
C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47-82. |
[16] |
Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. P.D.E., 23 (1998), 487-545.
doi: doi:10.1080/03605309808821354. |
[17] |
Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.
doi: doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z. |
[18] |
C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Diff. Eqns., 72 (1988), 1-27.
doi: doi:10.1016/0022-0396(88)90147-7. |
[19] |
F.-H. Lin, W.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.
doi: doi:10.1002/cpa.20139. |
[20] |
A. Malchiodi, Solutions concentrating at curves for some singularly perturbed elliptic problems, C. R. Math. Acad. Sci. Paris, 338 (2004), 775-780.
doi: doi:10.1016/j.crma.2004.03.023. |
[21] |
A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 55 (2002), 1507-1568.
doi: doi:10.1002/cpa.10049. |
[22] |
A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.
doi: doi:10.1215/S0012-7094-04-12414-5. |
[23] |
A. Malchiodi, W.-M. Ni and J. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 143-163. |
[24] |
W.-M. Ni and I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., 41 (1991), 819-851.
doi: doi:10.1002/cpa.3160440705. |
[25] |
W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
doi: doi:10.1215/S0012-7094-93-07004-4. |
[26] |
W.-M. Ni, I. Takagi and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems: intermediate solutions, Duke Math. J., 94 (1998), 597-618.
doi: doi:10.1215/S0012-7094-98-09424-8. |
[27] |
W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.
doi: doi:10.1002/cpa.3160480704. |
[28] |
Yang Wang, Concentration phenomena of solutions for some singularly perturbed elliptic equations, J. Math. Anal. Appl., 331 (2007) 927-946.
doi: doi:10.1016/j.jmaa.2006.09.029. |
[29] |
J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problems, J. Diff. Eqns., 129 (1996), 315-333.
doi: doi:10.1006/jdeq.1996.0120. |
[30] |
J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Diff. Eqns., 134 (1997), 104-133.
doi: doi:10.1006/jdeq.1996.3218. |
[31] |
J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J., 50 (1998), 159-178.
doi: doi:10.2748/tmj/1178224971. |
[32] |
J. Wei, On the effect of the domain geometry in singular perturbatation problems, Diff. Int. Eqns., 13 (2000), 15-45. |
[33] |
J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492. |
[34] |
J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. London Math. Soc., 59 (1999), 585-606.
doi: doi:10.1112/S002461079900719X. |
show all references
References:
[1] |
A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Comm. Math. Phys, 235 (2003), 427-466.
doi: doi:10.1007/s00220-003-0811-y. |
[2] |
A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part II, Indiana Univ. Math. J, 53 (2004), 297-329.
doi: doi:10.1512/iumj.2004.53.2400. |
[3] |
W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem, preprint 2010. |
[4] |
P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Diff. Eqns, 4 (1999), 1-69. |
[5] |
P. Bates and G. Fusco, Equilibria with many nuclei for the Cahn-Hilliard equation, J. Diff. Eqns, 160 (2000), 283-356.
doi: doi:10.1006/jdeq.1999.3660. |
[6] |
M. del Pino and D. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883-898.
doi: doi:10.1512/iumj.1999.48.1596. |
[7] |
M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79.
doi: doi:10.1137/S0036141098332834. |
[8] |
M. del Pino, P. Felmer and J. Wei, On the role of distance function in some singularly perturbed problems, Comm. P.D.E., 25 (2000), 155-177.
doi: doi:10.1080/03605300008821511. |
[9] |
M. del Pino, P. Felmer and J. Wei, Multiple-peak solutions for some singular perturbation problems, Cal. Var. P.D.E., 10 (2000), 119-134.
doi: doi:10.1007/s005260050147. |
[10] |
E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific. J. Math., 189 (1999), 241-262.
doi: doi:10.2140/pjm.1999.189.241. |
[11] |
B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^N$, in "Mathematical Analysis and Applications," Part A, Adv. Math. Suppl. Studies, Vol. 7A, pp.369-402, Academic Press, New York, 1981. |
[12] |
R. Gardner and L. A. Peletier, The set of positive solutions of semilinear equations in large balls, Proc. Roy. Soc. Edinburgh Sect. A, 104 (1986), 53-72. |
[13] |
C. Gui and J. Wei, Multiple interior spike solutions for some singularly perturbed Neumann problems, J. Diff. Eqns., 158 (1999), 1-27.
doi: doi:10.1016/S0022-0396(99)80016-3. |
[14] |
C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Can. J. Math., 52 (2000), 522-538.
doi: doi:10.4153/CJM-2000-024-x. |
[15] |
C. Gui, J. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47-82. |
[16] |
Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. P.D.E., 23 (1998), 487-545.
doi: doi:10.1080/03605309808821354. |
[17] |
Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.
doi: doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z. |
[18] |
C. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Diff. Eqns., 72 (1988), 1-27.
doi: doi:10.1016/0022-0396(88)90147-7. |
[19] |
F.-H. Lin, W.-M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.
doi: doi:10.1002/cpa.20139. |
[20] |
A. Malchiodi, Solutions concentrating at curves for some singularly perturbed elliptic problems, C. R. Math. Acad. Sci. Paris, 338 (2004), 775-780.
doi: doi:10.1016/j.crma.2004.03.023. |
[21] |
A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., 55 (2002), 1507-1568.
doi: doi:10.1002/cpa.10049. |
[22] |
A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.
doi: doi:10.1215/S0012-7094-04-12414-5. |
[23] |
A. Malchiodi, W.-M. Ni and J. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 143-163. |
[24] |
W.-M. Ni and I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., 41 (1991), 819-851.
doi: doi:10.1002/cpa.3160440705. |
[25] |
W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.
doi: doi:10.1215/S0012-7094-93-07004-4. |
[26] |
W.-M. Ni, I. Takagi and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems: intermediate solutions, Duke Math. J., 94 (1998), 597-618.
doi: doi:10.1215/S0012-7094-98-09424-8. |
[27] |
W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.
doi: doi:10.1002/cpa.3160480704. |
[28] |
Yang Wang, Concentration phenomena of solutions for some singularly perturbed elliptic equations, J. Math. Anal. Appl., 331 (2007) 927-946.
doi: doi:10.1016/j.jmaa.2006.09.029. |
[29] |
J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problems, J. Diff. Eqns., 129 (1996), 315-333.
doi: doi:10.1006/jdeq.1996.0120. |
[30] |
J. Wei, On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Diff. Eqns., 134 (1997), 104-133.
doi: doi:10.1006/jdeq.1996.3218. |
[31] |
J. Wei, On the interior spike layer solutions to a singularly perturbed Neumann problem, Tohoku Math. J., 50 (1998), 159-178.
doi: doi:10.2748/tmj/1178224971. |
[32] |
J. Wei, On the effect of the domain geometry in singular perturbatation problems, Diff. Int. Eqns., 13 (2000), 15-45. |
[33] |
J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 459-492. |
[34] |
J. Wei and M. Winter, Multi-peak solutions for a wide class of singular perturbation problems, J. London Math. Soc., 59 (1999), 585-606.
doi: doi:10.1112/S002461079900719X. |
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