Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon
Pages: 2411  2429,
Issue 6,
November
2015
doi:10.3934/cpaa.2015.14.2411 Abstract
References
Full text (484.6K)
Related Articles
Qiuping Lu  School of Mathematical Science, Yangzhou University, Yangzhou 225002, China, China (email)
Zhi Ling  School of Mathematical Science, Yangzhou University, Yangzhou 225002, China, China (email)
1 
A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\RR^N$, Progress in Mathematics, 240. Birkhauser Verlag, Basel, 2006. 

2 
A. Ambrosetti and P. H. Rabinowitz, Dual variational metheods in critical point theory and applications, J. Funct. Anal., 14 (1973), 341381. 

3 
H. Berestyki and P. L. Lions, Nonlinear scalar field equations I, Arch. Rat. Mech. Anal., 82 (1983), 313346. 

4 
J. Byeon, Singularly perturbed nonlinear Dirichlet problems with a general nonlinearity, Trans. Amer. Math. Soc., 362 (2010), 19812001. 

5 
D. Cao, N. E. Dancer, E. S. Noussair and S. Yan, On the existence and profile of multipeaked solutions to singularly perturbed semilinear Dirichlet problems, Discrete Contin. Dynam. Systems, 2 (1996), 221236. 

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), 211221. 

7 
C. Cortázar, M. Del Pino and M. Elgueta, Uniqueness and stability of regional blowup in a porousmedium equation, Ann. Inst. H. Poincaré Anal. Non Liné aire, 19 (2002), 927960. 

8 
C. Cortázar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in $\RR^N $ with a nonLipschitzian nonlinearity, Advances in Diff. Eqs., 1 (1996), 199218. 

9 
C. Cortázar, M. Elgueta and P. Felmer, Symmetry in an elliptic problem and the blowup set of a quasilinear heat equation, Comm. Partial Diff. Eqs., 21 (1996), 507520. 

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), 127141. 

11 
J. Dávila and M. Montenegro, Concentration for an elliptic equation with singular nonlinearity, J. Math. Pures Appl. (9), 97 (2012), 545578. 

12 
J. Dávila and M. Montenegro, Radial solutions of an elliptic equation with singular nonlinearity, J. Math. Anal. Appl., 352 (2009), 360379. 

13 
E. N. Dancer and S. Santra, Singular perturbed problems in the zero mass case: asymptotic behavior of spikes, Annali di Matematica, 189 (2010), 185225. 

14 
M. Del Pino and P. Felmer, Spikelayered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., 48 (1999), 883898. 

15 
M. Del Pino, P. Felmer and J. Wei, Multipeak solutions for some singular perturbation problems, Calc. Var. Partial Differential Equations, 10 (2000), 119134. 

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), 155177. 

17 
J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, Volume I, Elliptic Equations, Research Notes in Mathematics 106, Pitman, 1985. 

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), 283308. 

19 
M. Flucher and J. Wei, Asymptotic shape and location of small cores in elliptic freeboundary problems, Math. Z., 228 (1998), 683703. 

20 
V. A. Galaktionov, On a blowup set for the quasilinear heat equation $u_t=(u^{\sigma}u_x)_x+u^{\sigma+1}$, J. Differential Equations, 101 (1993), 6679. 

21 
B. Gidas W.M. Ni and L. Nirenberg, Symmetry and related properties via the Maximum Principle, Comm. Math. Phys., 68 (1979), 209243. 

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), 209243. 

23 
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer Verlag, New York, 1977. 

24 
C. Gui, Symmetry of the blowup set of a porous medium type equation, Comm. Pure Appl. Math., 48 (1995), 471500. 

25 
L. Jeanjean and K. Tanaka, A remark on the least energy solution in $\RR^N$, Proc. Amer. Math. Soc., 131 (2003), 23992408. 

26 
H. G. Kaper, M. K. Kwong and Y. Li, Symmetry results for reaction diffusion equations, Diff. Int. Eqs., 6 (1993), 10451056. 

27 
Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 14451490. 

28 
Q. Lu, Multiple solutions with compact support for a semilinear elliptic problem with critical growth, J. Differential Equations, 252 (2012), 62756305. 

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. 

30 
W.M. Ni and I. Takagi, On the shape of leastenergy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819851. 

31 
W.M. Ni and I. Takagi, Locating the peaks of leastenergy solutions to a semilinear Neumann problem, Duke Math. Journal, 70 (1993), 274281. 

32 
W.M. Ni and J. Wei, On the location and profile of spikelayer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731768. 

33 
E.S. Noussair and S. Yan, The effect of the domain geometry in singular perturbation problems, Proc. London Math. Soc. (3), 76 (1998), 427452. 

34 
L. A. Peletier and J. Serrin, Uniqueness of nonnegative solutions of semilinear equations in $\RR^N$, J. Diff. Eqs., 61 (1986), 380397. 

35 
P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270291. 

36 
J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897923. 

37 
M. Struwe, Variational Methods, SpringerVerlag, 1990. 

38 
J. Wei, On the construction of singlepeaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations, 129 (1996), 315333. 

Go to top
