# American Institute of Mathematical Sciences

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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Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. 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, J. Differential Equations, 129 (1996), 315-333. 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, 240. Birkhauser Verlag, Basel, 2006.  Google Scholar [2] A. Ambrosetti and P. H. Rabinowitz, Dual variational metheods in critical point theory and applications, J. Funct. Anal., 14 (1973), 341-381.  Google Scholar [3] H. Berestyki and P. L. Lions, Nonlinear scalar field equations I, Arch. Rat. Mech. Anal., 82 (1983), 313-346. Google Scholar [4] 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 [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, Discrete Contin. Dynam. Systems, 2 (1996), 221-236. 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, Comm. Math. Phys., 58 (1978), 211-221.  Google Scholar [7] C. Cortázar, M. Del Pino and M. Elgueta, Uniqueness and stability of regional blow-up in a porous-medium equation, Ann. Inst. H. Poincaré Anal. Non Liné aire, 19 (2002), 927-960. 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, Advances in Diff. Eqs., 1 (1996), 199-218.  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, Comm. Partial Diff. Eqs., 21 (1996), 507-520. 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 $\RR^N, N\ge3$, Arch. Rational Mech. Anal., 142 (1998), 127-141. doi: 10.1007/s002050050086.  Google Scholar [11] J. Dávila and M. Montenegro, Concentration for an elliptic equation with singular nonlinearity, J. Math. Pures Appl. (9), 97 (2012), 545-578. 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, J. Math. Anal. Appl., 352 (2009), 360-379. 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, Annali di Matematica, 189 (2010), 185-225. 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, Indiana Univ. Math. J., 48 (1999), 883-898. 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, Calc. Var. Partial Differential Equations, 10 (2000), 119-134. 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, Comm. Partial Differential Equations, 25 (2000), 155-177. doi: 10.1080/03605300008821511.  Google Scholar [17] J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Volume I, Elliptic Equations, Research Notes in Mathematics 106, Pitman, 1985.  Google Scholar [18] W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.  Google Scholar [19] M. Flucher and J. Wei, Asymptotic shape and location of small cores in elliptic free-boundary problems, Math. Z., 228 (1998), 683-703. doi: 10.1007/PL00004636.  Google Scholar [20] V. A. Galaktionov, On a blow-up set for the quasilinear heat equation $u_t=(u^{\sigma}u_x)_x+u^{\sigma+1}$, J. Differential Equations, 101 (1993), 66-79. 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, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar [22] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\RR^N$, Advances in Math. Studies, 7 A (1979), 209-243. Google Scholar [23] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, New York, 1977.  Google Scholar [24] C. Gui, Symmetry of the blow-up set of a porous medium type equation, Comm. Pure Appl. Math., 48 (1995), 471-500. doi: 10.1002/cpa.3160480502.  Google Scholar [25] L. Jeanjean and K. Tanaka, A remark on the least energy solution in $\RR^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. 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, Diff. Int. Eqs., 6 (1993), 1045-1056.  Google Scholar [27] 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 [28] Q. Lu, Multiple solutions with compact support for a semilinear elliptic problem with critical growth, J. Differential Equations, 252 (2012), 6275-6305. 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, Comm. Pure Appl. Math., 44 (1991), 819-851. 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, Duke Math. Journal, 70 (1993), 274-281. 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, Comm. Pure Appl. Math., 48 (1995), 731-768. doi: 10.1002/cpa.3160480704.  Google Scholar [33] E.S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc. (3), 76 (1998), 427-452. doi: 10.1112/S0024611598000148.  Google Scholar [34] L. A. Peletier and J. Serrin, Uniqueness of non-negative solutions of semilinear equations in $\RR^N$, J. Diff. Eqs., 61 (1986), 380-397. doi: 10.1016/0022-0396(86)90112-9.  Google Scholar [35] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631.  Google Scholar [36] J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923. 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, J. Differential Equations, 129 (1996), 315-333. doi: 10.1006/jdeq.1996.0120.  Google Scholar
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