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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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

show all references

References:
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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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