Multibump solutions for an elliptic problem in expanding domains.(English)Zbl 1011.35059

Let $$\Omega$$ be a bounded convex domain in $$\mathbb{R}^N$$ with $$C^2$$ boundary $$\partial\Omega$$ and $$\nu(x)$$ be the outward unit normal of $$\partial \Omega$$ at $$x$$. Define $\Omega_R= \bigl\{x\mid x=\widetilde x+t\nu(\widetilde x),\;\forall\widetilde x\in\partial\Omega,\;t\in[R,R+1] \bigr\}.$ The aim of this paper is to construct multibump solutions for an elliptic problem on this expanding domain, such that all the local maximum points of the solution are close to the set $$\{x\mid x=\widetilde x+(R+1/2) \nu(\widetilde x)$$, $$\widetilde x\in \partial\Omega\}$$ if $$R>0$$ is large enough.

MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

convex domain; local maximum
Full Text:

References:

 [1] Bahri A., Research Notes in Mathematics 182 (1989) [2] DOI: 10.1006/jdeq.1996.3241 · Zbl 0878.35043 [3] DOI: 10.3934/dcds.1996.2.221 · Zbl 0947.35073 [4] DOI: 10.1006/jdeq.1998.3600 · Zbl 0944.35026 [5] DOI: 10.1016/0022-0396(84)90153-0 · Zbl 0569.35033 [6] DOI: 10.1007/BF01452052 · Zbl 0699.35103 [7] Dancer E.N., Top. Methods Nonlinear Anal. 11 pp 227– (1998) [8] DOI: 10.2140/pjm.1999.189.241 · Zbl 0933.35070 [9] Dancer E.N., Adv. Diff. Equations 4 pp 347– (1999) [10] Dancer E.N., J. Diff. Int. Equations 13 pp 747– (2000) [11] Dancer E.N., Indiana University J. Math. 48 pp 1177– (1999) · Zbl 0948.35055 [12] Dold A., Lectures on Algebraic Topology (1972) · Zbl 0234.55001 [13] Gidas B., Mathematical Anal. Appl., Adv. Math. Supplementary Studies 7 pp 369– (1981) [14] Gilbarg D., Elliptic Partial Differential Equations of Second Order (1983) · Zbl 0562.35001 [15] Grossi M., Adv. Diff. Equations 5 pp 193– (2000) [16] Hille E., Lectures on Ordinary Differential Equations (1969) · Zbl 0179.40301 [17] DOI: 10.1112/jlms/s2-31.3.566 · Zbl 0573.58007 [18] Kwong M.H., Diff. Int. Equations 4 pp 583– (1991) [19] DOI: 10.1016/0022-0396(90)90062-T · Zbl 0748.35013 [20] DOI: 10.1006/jdeq.1993.1053 · Zbl 0803.35053 [21] DOI: 10.1006/jdeq.1995.1112 · Zbl 0839.35039 [22] Mizoguchi N., Houston J. Math. 1 pp 199– (1996) [23] DOI: 10.1215/S0012-7094-93-07004-4 · Zbl 0796.35056 [24] DOI: 10.1002/cpa.3160480704 · Zbl 0838.35009 [25] DOI: 10.1016/0022-1236(90)90002-3 · Zbl 0786.35059 [26] DOI: 10.1080/03605307908820096 · Zbl 0462.35016
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