Blow-up rates of radially symmetric large solutions.(English)Zbl 1163.35015

Summary: This paper adapts a technical device going back to [J. López-Gómez, J. Differ. Equations 224, No. 2, 385–439 (2006; Zbl 1208.35036)] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Discrete Contin. Dyn. Syst., Suppl. 2007, 677–686 (2007; Zbl 1163.35352)]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova and J. López-Gómez, J. Differ. Equations 244, No. 12, 3180–3203 (2008; Zbl 1149.34020)]. Precisely, we show that if $$\Omega$$ is a ball, or an annulus, $$f\in{\mathcal C}[0,\infty)$$ is positive and non-decreasing, $$V\in {\mathcal C}[0,\infty)\cap{\mathcal C}^2(0,\infty)$$ satisfies $$V(0)=0$$, $$V'(u)>0$$, $$V''(u)\geq 0$$, for every $$u>0$$, and $$V(u)\sim Hu^{p-1}$$ as $$u\uparrow\infty$$, for some $$H>0$$ and $$p>1$$, then, for each $$\lambda\geq 0$$,
$-\Delta u=\lambda u- f\big(\text{dist}(x,\partial\Omega)\big)V(u)u$
possesses a unique positive large solution in $$\Omega$$, $$L$$, which must be radially symmetric, by uniqueness, and we can estimate the exact blow-up rate of $$L(x)$$ at $$\partial\Omega$$ in terms of $$p$$, $$H$$ and $$f$$.

MSC:

 35J60 Nonlinear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs
Full Text:

References:

 [1] Cano-Casanova, S.; López-Gómez, J., Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. differential equations, 244, 3180-3203, (2008) · Zbl 1149.34020 [2] Cirstea, F.C.; Radulescu, V., Uniqueness of the blow-up boundary solution of logistic equations with adsorption, C. R. acad. sci. Paris ser. I, 335, 447-452, (2002) · Zbl 1183.35124 [3] Cirstea, F.C.; Radulescu, V., Asymptotics for the blow-up boundary solution of the logistic equations with adsorption, C. R. acad. sci. Paris ser. I, 336, 231-236, (2003) · Zbl 1068.35035 [4] Chuaqui, M.; Cortázar, C.; Elgueta, M.; García-Melián, J., Uniqueness and boundary behaviour of large solutions to elliptic problems with singular weights, Commun. pure appl. anal., 3, 653-662, (2004) · Zbl 1174.35386 [5] Chuaqui, M.; Cortázar, C.; Elgueta, M.; Flores, C.; García-Melián, J., On an elliptic problem with boundary blow-up and a singular weight: the radial case, Proc. roy. soc. Edinburgh sect. A, 133A, 1283-1297, (2003) · Zbl 1039.35036 [6] Du, Y.; Huang, Q., Blow-up solutions for a class of semilinear elliptic and parabolic problems, SIAM J. math. anal., 31, 1-18, (1999) · Zbl 0959.35065 [7] García-Melián, J., Uniqueness for boundary blow-up problems with continuous weights, Proc. amer. math. soc., 135, 2785-2793, (2007) · Zbl 1146.35036 [8] García-Melián, J., Boundary behavior for large solutions of elliptic equations with singular weights, Nonlinear anal., 67, 818-826, (2007) · Zbl 1143.35054 [9] García-Melián, J.; Gómez-Reñasco, R.; López-Gómez, J.; Sabina de Lis, J.C., Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. ration. mech. anal., 145, 261-289, (1998) · Zbl 0926.35036 [10] García-Melián, J.; Letelier-Albornoz, R.; Sabina de Lis, J.C., Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. amer. math. soc., 129, 3593-3602, (2001) · Zbl 0989.35044 [11] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1977), Springer-Verlag Berlin · Zbl 0691.35001 [12] R. Gómez-Reñasco, The effect of varying coefficients in semilinear elliptic boundary value problems. From classical solutions to metasolutions, Ph.D. Thesis, La Laguna University (Tenerife), February 1999 [13] Gómez-Reñasco, R.; López-Gómez, J., On the existence and numerical computation of classical and non-classical solutions for a family of elliptic boundary value problems, Nonlinear anal., 48, 567-605, (2002) · Zbl 1113.35079 [14] Keller, J.B., On solutions of $$\Delta u = f(u)$$, Comm. pure appl. math., X, 503-510, (1957) · Zbl 0090.31801 [15] López-Gómez, J., Large solutions, metasolutions, and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems, Electron. J. differ. equ. conf., 05, 135-171, (2000) · Zbl 1055.35049 [16] López-Gómez, J., The boundary blow-up rate of large solutions, J. differential equations, 195, 25-45, (2003) · Zbl 1130.35329 [17] López-Gómez, J., Dynamics of parabolic equations: from classical solutions to metasolutions, Differential integral equations, 16, 813-828, (2003) · Zbl 1036.35080 [18] López-Gómez, J., Metasolutions: malthus versus verhulst in population dynamics. A dream of Volterra, (), 211-309 · Zbl 1102.35001 [19] López-Gómez, J., Optimal uniqueness theorems and exact blow-up rates of large solutions, J. differential equations, 224, 385-439, (2006) · Zbl 1208.35036 [20] López-Gómez, J., Uniqueness of large solutions for a class of radially symmetric elliptic equations, (), 75-110 · Zbl 1133.35359 [21] J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677-686 [22] Osserman, R., On the inequality $$\Delta u \geqslant f(u)$$, Pacific J. math., 7, 1641-1647, (1957) · Zbl 0083.09402 [23] Ouyang, T.; Xie, Z., The uniqueness of blow-up for radially symmetric semilinear elliptic equations, Nonlinear anal., 64, 2129-2142, (2006) · Zbl 1161.35005 [24] Ouyang, T.; Xie, Z., The exact boundary blow-up rate of large solutions for semilinear elliptic problems, Nonlinear anal., 68, 2791-2800, (2008) · Zbl 1138.35025 [25] Radulescu, V., Singular phenomena in nonlinear elliptic problems: from boundary blow-up solutions to equations with singular nonlinearities, (), 483-591
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