November  2015, 14(6): 2411-2429. doi: 10.3934/cpaa.2015.14.2411

Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon

1. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Received  March 2015 Revised  July 2015 Published  September 2015

For a general elliptic problem $-\triangle{u} = g(u)$ in $R^N$ with $N \ge 3$, we show that all solutions have compact support and there exists a least energy solution, which is radially symmetric and decreases with respect to $|x| = r$. With this result we study a singularly perturbed elliptic problem $ -\epsilon^{2} \triangle{u} + |u|^{q-1}u = \lambda u + f(u)$ in a bounded domain $\Omega$ with $0 < q < 1 $, $\lambda \ge 0$ and $ u \in H^1_0(\Omega) $. For any $y \in \Omega$, we show that there exists a least energy solution $u_{\epsilon}$, which concentrates around this point $y$ as $\epsilon \to 0$. Conversely when $\epsilon $ is small, the boundary of the set $\{ x \in \Omega | u_{\epsilon}(x)>0 \}$ is a free boundary, where $u_{\epsilon}$ is any nonnegative least energy solution.
Citation: Qiuping Lu, Zhi Ling. Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2411-2429. doi: 10.3934/cpaa.2015.14.2411
References:
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A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\RR^N$,, Progress in Mathematics, (2006).   Google Scholar

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A. Ambrosetti and P. H. Rabinowitz, Dual variational metheods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 341.   Google Scholar

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S. Coleman, V. Glaser and A. Martin, Action minima among solutions to a class of Euclidean scalar field equation,, \emph{Comm. Math. Phys.}, 58 (1978), 211.   Google Scholar

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C. Cortázar, M. Del Pino and M. Elgueta, Uniqueness and stability of regional blow-up in a porous-medium equation,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'e aire}, 19 (2002), 927.  doi: 10.1016/S0294-1449(02)00107-5.  Google Scholar

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C. Cortázar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in $\RR^N $ with a non-Lipschitzian non-linearity,, \emph{Advances in Diff. Eqs.}, 1 (1996), 199.   Google Scholar

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C. Cortázar, M. Elgueta and P. Felmer, Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation,, \emph{Comm. Partial Diff. Eqs.}, 21 (1996), 507.  doi: 10.1080/03605309608821194.  Google Scholar

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C. Cortázar, M. Elgueta and P. Felmer, Uniqueness of positive solutions of $\Delta u+f(u)=0$ in $\RR^N, N\ge3$,, \emph{Arch. Rational Mech. Anal.}, 142 (1998), 127.  doi: 10.1007/s002050050086.  Google Scholar

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J. Dávila and M. Montenegro, Concentration for an elliptic equation with singular nonlinearity,, \emph{J. Math. Pures Appl. (9)}, 97 (2012), 545.  doi: 10.1016/j.matpur.2011.02.001.  Google Scholar

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E. N. Dancer and S. Santra, Singular perturbed problems in the zero mass case: asymptotic behavior of spikes,, \emph{Annali di Matematica}, 189 (2010), 185.  doi: 10.1007/s10231-009-0105-x.  Google Scholar

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

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W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation,, \emph{Arch. Rational Mech. Anal.}, 91 (1986), 283.  doi: 10.1007/BF00282336.  Google Scholar

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M. Flucher and J. Wei, Asymptotic shape and location of small cores in elliptic free-boundary problems,, \emph{Math. Z.}, 228 (1998), 683.  doi: 10.1007/PL00004636.  Google Scholar

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B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\RR^N$,, \emph{Advances in Math. Studies}, 7 A (1979), 209.   Google Scholar

[23]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Springer Verlag, (1977).   Google Scholar

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C. Gui, Symmetry of the blow-up set of a porous medium type equation,, \emph{Comm. Pure Appl. Math.}, 48 (1995), 471.  doi: 10.1002/cpa.3160480502.  Google Scholar

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L. Jeanjean and K. Tanaka, A remark on the least energy solution in $\RR^N$,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 2399.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

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H. G. Kaper, M. K. Kwong and Y. Li, Symmetry results for reaction diffusion equations,, \emph{Diff. Int. Eqs.}, 6 (1993), 1045.   Google Scholar

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

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Q. Lu, Multiple solutions with compact support for a semilinear elliptic problem with critical growth,, \emph{J. Differential Equations}, 252 (2012), 6275.  doi: 10.1016/j.jde.2012.03.001.  Google Scholar

[29]

Q. Lu, Locating the peaks of the least energy solutions to an elliptic problem involving sublinear term with Neumann boundary condition,, Work in progress., ().   Google Scholar

[30]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, \emph{Comm. Pure Appl. Math.}, 44 (1991), 819.  doi: 10.1002/cpa.3160440705.  Google Scholar

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W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, \emph{Duke Math. Journal}, 70 (1993), 274.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[32]

W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, \emph{Comm. Pure Appl. Math.}, 48 (1995), 731.  doi: 10.1002/cpa.3160480704.  Google Scholar

[33]

E.S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems,, \emph{Proc. London Math. Soc. (3)}, 76 (1998), 427.  doi: 10.1112/S0024611598000148.  Google Scholar

[34]

L. A. Peletier and J. Serrin, Uniqueness of non-negative solutions of semilinear equations in $\RR^N$,, \emph{J. Diff. Eqs.}, 61 (1986), 380.  doi: 10.1016/0022-0396(86)90112-9.  Google Scholar

[35]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270.  doi: 10.1007/BF00946631.  Google Scholar

[36]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations,, \emph{Indiana Univ. Math. J.}, 49 (2000), 897.  doi: 10.1512/iumj.2000.49.1893.  Google Scholar

[37]

M. Struwe, Variational Methods,, Springer-Verlag, (1990).  doi: 10.1007/978-3-662-02624-3.  Google Scholar

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

show all references

References:
[1]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\RR^N$,, Progress in Mathematics, (2006).   Google Scholar

[2]

A. Ambrosetti and P. H. Rabinowitz, Dual variational metheods in critical point theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 341.   Google Scholar

[3]

H. Berestyki and P. L. Lions, Nonlinear scalar field equations I,, \emph{Arch. Rat. Mech. Anal.}, 82 (1983), 313.   Google Scholar

[4]

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

[5]

D. Cao, N. E. Dancer, E. S. Noussair and S. Yan, On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems,, \emph{Discrete Contin. Dynam. Systems}, 2 (1996), 221.  doi: 10.3934/dcds.1996.2.221.  Google Scholar

[6]

S. Coleman, V. Glaser and A. Martin, Action minima among solutions to a class of Euclidean scalar field equation,, \emph{Comm. Math. Phys.}, 58 (1978), 211.   Google Scholar

[7]

C. Cortázar, M. Del Pino and M. Elgueta, Uniqueness and stability of regional blow-up in a porous-medium equation,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'e aire}, 19 (2002), 927.  doi: 10.1016/S0294-1449(02)00107-5.  Google Scholar

[8]

C. Cortázar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in $\RR^N $ with a non-Lipschitzian non-linearity,, \emph{Advances in Diff. Eqs.}, 1 (1996), 199.   Google Scholar

[9]

C. Cortázar, M. Elgueta and P. Felmer, Symmetry in an elliptic problem and the blow-up set of a quasilinear heat equation,, \emph{Comm. Partial Diff. Eqs.}, 21 (1996), 507.  doi: 10.1080/03605309608821194.  Google Scholar

[10]

C. Cortázar, M. Elgueta and P. Felmer, Uniqueness of positive solutions of $\Delta u+f(u)=0$ in $\RR^N, N\ge3$,, \emph{Arch. Rational Mech. Anal.}, 142 (1998), 127.  doi: 10.1007/s002050050086.  Google Scholar

[11]

J. Dávila and M. Montenegro, Concentration for an elliptic equation with singular nonlinearity,, \emph{J. Math. Pures Appl. (9)}, 97 (2012), 545.  doi: 10.1016/j.matpur.2011.02.001.  Google Scholar

[12]

J. Dávila and M. Montenegro, Radial solutions of an elliptic equation with singular nonlinearity,, \emph{J. Math. Anal. Appl.}, 352 (2009), 360.  doi: 10.1016/j.jmaa.2008.05.033.  Google Scholar

[13]

E. N. Dancer and S. Santra, Singular perturbed problems in the zero mass case: asymptotic behavior of spikes,, \emph{Annali di Matematica}, 189 (2010), 185.  doi: 10.1007/s10231-009-0105-x.  Google Scholar

[14]

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

[15]

M. Del Pino, P. Felmer and J. Wei, Multi-peak solutions for some singular perturbation problems,, \emph{Calc. Var. Partial Differential Equations}, 10 (2000), 119.  doi: 10.1007/s005260050147.  Google Scholar

[16]

M. Del Pino, P. Felmer and J. Wei, On the role of distance function in some singular perturbation problems,, \emph{Comm. Partial Differential Equations}, 25 (2000), 155.  doi: 10.1080/03605300008821511.  Google Scholar

[17]

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries,, Volume I, (1985).   Google Scholar

[18]

W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation,, \emph{Arch. Rational Mech. Anal.}, 91 (1986), 283.  doi: 10.1007/BF00282336.  Google Scholar

[19]

M. Flucher and J. Wei, Asymptotic shape and location of small cores in elliptic free-boundary problems,, \emph{Math. Z.}, 228 (1998), 683.  doi: 10.1007/PL00004636.  Google Scholar

[20]

V. A. Galaktionov, On a blow-up set for the quasilinear heat equation $u_t=(u^{\sigma}u_x)_x+u^{\sigma+1}$,, \emph{J. Differential Equations}, 101 (1993), 66.  doi: 10.1006/jdeq.1993.1005.  Google Scholar

[21]

B. Gidas W.-M. Ni and L. Nirenberg, Symmetry and related properties via the Maximum Principle,, \emph{Comm. Math. Phys.}, 68 (1979), 209.   Google Scholar

[22]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\RR^N$,, \emph{Advances in Math. Studies}, 7 A (1979), 209.   Google Scholar

[23]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Springer Verlag, (1977).   Google Scholar

[24]

C. Gui, Symmetry of the blow-up set of a porous medium type equation,, \emph{Comm. Pure Appl. Math.}, 48 (1995), 471.  doi: 10.1002/cpa.3160480502.  Google Scholar

[25]

L. Jeanjean and K. Tanaka, A remark on the least energy solution in $\RR^N$,, \emph{Proc. Amer. Math. Soc.}, 131 (2003), 2399.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar

[26]

H. G. Kaper, M. K. Kwong and Y. Li, Symmetry results for reaction diffusion equations,, \emph{Diff. Int. Eqs.}, 6 (1993), 1045.   Google Scholar

[27]

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

[28]

Q. Lu, Multiple solutions with compact support for a semilinear elliptic problem with critical growth,, \emph{J. Differential Equations}, 252 (2012), 6275.  doi: 10.1016/j.jde.2012.03.001.  Google Scholar

[29]

Q. Lu, Locating the peaks of the least energy solutions to an elliptic problem involving sublinear term with Neumann boundary condition,, Work in progress., ().   Google Scholar

[30]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, \emph{Comm. Pure Appl. Math.}, 44 (1991), 819.  doi: 10.1002/cpa.3160440705.  Google Scholar

[31]

W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, \emph{Duke Math. Journal}, 70 (1993), 274.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[32]

W.-M. Ni and J. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,, \emph{Comm. Pure Appl. Math.}, 48 (1995), 731.  doi: 10.1002/cpa.3160480704.  Google Scholar

[33]

E.S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems,, \emph{Proc. London Math. Soc. (3)}, 76 (1998), 427.  doi: 10.1112/S0024611598000148.  Google Scholar

[34]

L. A. Peletier and J. Serrin, Uniqueness of non-negative solutions of semilinear equations in $\RR^N$,, \emph{J. Diff. Eqs.}, 61 (1986), 380.  doi: 10.1016/0022-0396(86)90112-9.  Google Scholar

[35]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270.  doi: 10.1007/BF00946631.  Google Scholar

[36]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations,, \emph{Indiana Univ. Math. J.}, 49 (2000), 897.  doi: 10.1512/iumj.2000.49.1893.  Google Scholar

[37]

M. Struwe, Variational Methods,, Springer-Verlag, (1990).  doi: 10.1007/978-3-662-02624-3.  Google Scholar

[38]

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

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