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March  2017, 16(2): 671-698. doi: 10.3934/cpaa.2017033

A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential

2-32-2, Kami-takada, Nakano, Tokyo 164-0002, Japan

Received  April 2016 Revised  August 2016 Published  January 2017

In this paper we study the following non-autonomous singularly perturbed Dirichlet problem:
${\varepsilon ^2}\Delta u -u + K(x)f(u) = 0, \; u > 0\quad {\rm{in}}\; \Omega, \quad u = 0\quad {\rm{on}}\; \partial \Omega, $
for a totally degenerate potential K. Here ε > 0 is a small parameter, $\Omega \subset \mathbb{R}^N$ is a bounded domain with a smooth boundary, and f is an appropriate superlinear subcritical function. In particular, f satisfies $0 < \liminf_{ t \to 0+} f(t)/t^q \leq \limsup_{ t \to 0+} f(t)/t^q < + \infty$ for some $1 < q < + \infty$. We show that the least energy solutions concentrate at the maximal point of the modified distance function $D(x) = \min \{ (q+1) d(x, \partial A), 2 d(x, \partial \Omega) \}$, where $A = \{ x \in \bar{ \Omega } \mid K(x) = \max_{ y \in \bar{ \Omega } } K(y) \}$ is assumed to be a totally degenerate set satisfying ${{A}^{{}^\circ }}\ne \emptyset $.
Citation: Shun Kodama. A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential. Communications on Pure & Applied Analysis, 2017, 16 (2) : 671-698. doi: 10.3934/cpaa.2017033
References:
[1]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Existence Results via the Variational Approach, Universitext, Springer, London, 2011. doi: 10.1007/978-0-85729-227-8.  Google Scholar

[2]

H. BerestyckiT. Gallouët and O. Kavian, Equations de champs scalaires euclidiens nonlinéaires dans le plan, C. R. Acad. Sc. Paris, Série I Math., 297 (1983), 307-310.   Google Scholar

[3]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-375.  doi: 10.1007/BF00250555.  Google Scholar

[4]

J. Byeon, Mountain pass solutions for singularly perturbed nonlinear Dirichlet problems, J. Differential Equations, 217 (2005), 257-281.  doi: 10.1016/j.jde.2005.07.008.  Google Scholar

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J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc., 362 (2010), 1981-2001.  doi: 10.1090/S0002-9947-09-04746-1.  Google Scholar

[6]

D. CaoN. DancerE. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete Contin. Dynam. Systems, 2 (1996), 221-236.  doi: 10.3934/dcds.1996.2.221.  Google Scholar

[7]

M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. math. J., 48 (1999), 883-898.  doi: 10.1512/iumj.1999.48.1596.  Google Scholar

[8]

M. del PinoP. Felmer and J. Wei, Multi-peak solutions for some singular perturbation problems, Calc. Var. Partial Differential Equations, 10 (2000), 119-134.  doi: 10.1007/s005260050147.  Google Scholar

[9]

B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Advances in Math., Supplementary Studies, 7A (1981), 369-402.   Google Scholar

[10]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Second, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1(2011).  Google Scholar

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.   Google Scholar

[12]

S. Kesavan, Symmetrization & Applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd. , Hackensack, NJ, 2006. doi: 10.1142/9789812773937.  Google Scholar

[13]

S. Kodama, On a concentration phenomenon of the least energy solution to nonlinear Schrödinger equations with a totally degenerate potential, preprint. Google Scholar

[14]

Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.3.CO;2-Q.  Google Scholar

[15]

C. S. LinW. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[16]

G. Lu and J. Wei, On nonlinear Schrödinger equations with totally degenerate potentials, C. R. Acad. Sci. Paris Sr. I Math., 326 (1998), 691-696.  doi: 10.1016/S0764-4442(98)80032-3.  Google Scholar

[17]

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: 10.1002/cpa.3160480704.  Google Scholar

[18]

N. Qiao and Q. Z.-Wang, Multiplicity results for positive solutions to non-autonomous elliptic problems, Electron. J. Differential Equations, 28 (1999), 1-28.   Google Scholar

[19]

X. Ren, Least-energy solutions to a non-autonomous semilinear problem with small diffusion coefficient, Electronic J. Partial Differential Equations, 5 (1993), 1-21.   Google Scholar

[20] G. N. Watson, A Treatise on the Theory of Bessel Functions. Reprint of the second (1944) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.   Google Scholar
[21]

J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations, 129 (1996), 315-333.  doi: 10.1006/jdeq.1996.0120.  Google Scholar

[22]

C. Zhao, On the number of interior peaks of solutions to a non-autonomous singularly perturbed Neumann problem, Proc. Roy. Soc. Edinburgh Sect., 139 (2009), 427-448.  doi: 10.1017/S0308210507001229.  Google Scholar

show all references

References:
[1]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Existence Results via the Variational Approach, Universitext, Springer, London, 2011. doi: 10.1007/978-0-85729-227-8.  Google Scholar

[2]

H. BerestyckiT. Gallouët and O. Kavian, Equations de champs scalaires euclidiens nonlinéaires dans le plan, C. R. Acad. Sc. Paris, Série I Math., 297 (1983), 307-310.   Google Scholar

[3]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-375.  doi: 10.1007/BF00250555.  Google Scholar

[4]

J. Byeon, Mountain pass solutions for singularly perturbed nonlinear Dirichlet problems, J. Differential Equations, 217 (2005), 257-281.  doi: 10.1016/j.jde.2005.07.008.  Google Scholar

[5]

J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc., 362 (2010), 1981-2001.  doi: 10.1090/S0002-9947-09-04746-1.  Google Scholar

[6]

D. CaoN. DancerE. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete Contin. Dynam. Systems, 2 (1996), 221-236.  doi: 10.3934/dcds.1996.2.221.  Google Scholar

[7]

M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. math. J., 48 (1999), 883-898.  doi: 10.1512/iumj.1999.48.1596.  Google Scholar

[8]

M. del PinoP. Felmer and J. Wei, Multi-peak solutions for some singular perturbation problems, Calc. Var. Partial Differential Equations, 10 (2000), 119-134.  doi: 10.1007/s005260050147.  Google Scholar

[9]

B. GidasW. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Advances in Math., Supplementary Studies, 7A (1981), 369-402.   Google Scholar

[10]

Q. Han and F. Lin, Elliptic Partial Differential Equations, Second, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1(2011).  Google Scholar

[11]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.   Google Scholar

[12]

S. Kesavan, Symmetrization & Applications, Series in Analysis, 3, World Scientific Publishing Co. Pte. Ltd. , Hackensack, NJ, 2006. doi: 10.1142/9789812773937.  Google Scholar

[13]

S. Kodama, On a concentration phenomenon of the least energy solution to nonlinear Schrödinger equations with a totally degenerate potential, preprint. Google Scholar

[14]

Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.3.CO;2-Q.  Google Scholar

[15]

C. S. LinW. M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[16]

G. Lu and J. Wei, On nonlinear Schrödinger equations with totally degenerate potentials, C. R. Acad. Sci. Paris Sr. I Math., 326 (1998), 691-696.  doi: 10.1016/S0764-4442(98)80032-3.  Google Scholar

[17]

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: 10.1002/cpa.3160480704.  Google Scholar

[18]

N. Qiao and Q. Z.-Wang, Multiplicity results for positive solutions to non-autonomous elliptic problems, Electron. J. Differential Equations, 28 (1999), 1-28.   Google Scholar

[19]

X. Ren, Least-energy solutions to a non-autonomous semilinear problem with small diffusion coefficient, Electronic J. Partial Differential Equations, 5 (1993), 1-21.   Google Scholar

[20] G. N. Watson, A Treatise on the Theory of Bessel Functions. Reprint of the second (1944) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.   Google Scholar
[21]

J. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations, 129 (1996), 315-333.  doi: 10.1006/jdeq.1996.0120.  Google Scholar

[22]

C. Zhao, On the number of interior peaks of solutions to a non-autonomous singularly perturbed Neumann problem, Proc. Roy. Soc. Edinburgh Sect., 139 (2009), 427-448.  doi: 10.1017/S0308210507001229.  Google Scholar

Figure 1.  Example 2 (ⅰ)
Figure 2.  Example 2 (ⅱ)
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