May  2019, 39(5): 2555-2579. doi: 10.3934/dcds.2019107

Positive radial solutions involving nonlinearities with zeros

1. 

Departamento de Matemática, Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso, Chile

2. 

Dipartimento di Scienze Matematiche e Ingegneria Industriale, Università Politecnica delle Marche, Via Brecce Bianche 1, 60131 Ancona, Italy

3. 

Departamento de Matemática, Universidad Técnica Federico Santa María, Av. España 1680, Valparaíso, Chile

Received  April 2018 Revised  October 2018 Published  January 2019

In this paper we consider the non-autonomous quasilinear elliptic problem
$ \begin{cases} -\Delta_p u = \lambda |x|^{\delta} f(u) &\mbox{in }B_1(0)\\ u = 0 &\mbox{in }\partial B_1(0), \end{cases} $
where
$ f:\mathbb{R}\to[0,\infty) $
is a nonnegative
$ C^1- $
function with
$ f(0) = 0 $
,
$ f(U) = 0 $
for some
$ U>0 $
, and
$ f $
is superlinear in
$ 0 $
and in
$ U $
. Assuming subcriticality either in
$ U $
or at infinity we study existence and multiplicity of positive radial solutions with respect to the parameter
$ \lambda $
. In addition, we study the bifurcation diagrams with respect to the maximum over the eventual solutions as the parameter
$ \lambda $
varies in the positive halfline.
Citation: Isabel Flores, Matteo Franca, Leonelo Iturriaga. Positive radial solutions involving nonlinearities with zeros. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2555-2579. doi: 10.3934/dcds.2019107
References:
[1]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[2]

A. AmbrosettiJ. Garcia Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242.  doi: 10.1006/jfan.1996.0045.

[3]

B. BarriosJ. Garca-Melin and L. Iturriaga, Semilinear elliptic equations and nonlinearities with zeros, Nonlinear Anal., 134 (2016), 117-126.  doi: 10.1016/j.na.2015.12.025.

[4]

M. F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324.  doi: 10.1007/BF00251552.

[5]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

[6]

F. Dalbono and M. Franca, Nodal solutions for supercritical Laplace equations, Commun in Math. Phys., 347 (2016), 875-901.  doi: 10.1007/s00220-015-2546-y.

[7]

D. G. De FigueiredoJ. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.

[8]

D. G. de FigueiredoP.-L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63. 

[9]

O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005.

[10]

I. Flores and M. Franca, Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball, Nonlinear Anal., 125 (2015), 128-149.  doi: 10.1016/j.na.2015.04.015.

[11]

R. H. Fowler, Further studies of Emden's and similar differential equations, Quart. Jl. Math., 2 (1931), 259-288.  doi: 10.1093/qmath/os-2.1.259.

[12]

M. Franca, Classification of positive solution of $p$-Laplace equation with a growth term, Arch. Math. (Brno), 40 (2004), 415-434. 

[13]

M. Franca, A dynamical approach to the study of radial solutions for $p$-Laplace equation, Rend. Semin. Mat. Univ. Politec. Torino, 65 (2007), 53-88. 

[14]

M. Franca, Radial ground states and singular ground states for a spatial dependent $p$-Laplace equation, J. Differential Equations, 248 (2010), 2629-2656.  doi: 10.1016/j.jde.2010.02.012.

[15]

M. Franca, Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type, Ann. Mat. Pura Appl., 189 (2010), 67-94.  doi: 10.1007/s10231-009-0101-1.

[16]

M. Franca and A. Sfecci, Entire solutions of superlinear problems with indefinite weights and Hardy potentials, J. Dyn. Differential Equations, 30 (2018), 1081-1118.  doi: 10.1007/s10884-017-9589-z.

[17]

B. FranchiE. Lanconelli and J. Serrin, Existence and uniqueness of nonnegative solutions of quasilinear equations in ${\mathbb{R}}^n$, Adv. in Math., 118 (1996), 177-243.  doi: 10.1006/aima.1996.0021.

[18]

J. Garca-Melin and L. Iturriaga, Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros, Israel J. Math., 210 (2015), 233-344.  doi: 10.1007/s11856-015-1251-z.

[19]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astron. and Astroph., 24 (1973), 229-238. 

[20]

L. IturriagaS. Lorca and E. Massa, Positive solutions for the $p$-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. H. Poincare Anal. Non Linaire, 27 (2010), 763-771.  doi: 10.1016/j.anihpc.2009.11.003.

[21]

L. IturriagaS. Lorca and E. Massa, Multiple positive solutions for the m-Laplacian and a nonlinearity with many zeros, Differential Integral Equations, 30 (2017), 145-159. 

[22]

L. IturriagaE. MassaJ. Snchez and P. Ubilla, Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327.  doi: 10.1016/j.jde.2009.08.008.

[23]

R. JohnsonX. B. Pan and Y. F. Yi, Singular ground states of semilinear elliptic equations via invariant manifold theory, Nonlinear Anal., 20 (1993), 1279-1302.  doi: 10.1016/0362-546X(93)90132-C.

[24]

R. JohnsonX. B. Pan and Y. F. Yi, The Melnikov method and elliptic equation with critical exponent, Indiana Univ. Math. J., 43 (1994), 1045-1077.  doi: 10.1512/iumj.1994.43.43046.

[25]

N. KawanoW. M. Ni and S. Yotsutani, A generalized Pohozaev identity and its applications, J. Math. Soc. Japan, 42 (1990), 541-564.  doi: 10.2969/jmsj/04230541.

[26]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.

[27]

Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398.  doi: 10.1006/jfan.1999.3446.

[28]

W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.  doi: 10.1512/iumj.1982.31.31056.

[29]

S. I. Pohozaev, Eigenfunctions of the equations $ \Delta u+ \lambda f(u) = 0$, Soviet Math. Dokl., 165 (1965), 1408-1411. 

[30]

S. Prashanth and K. Sreenadh, Multiplicity results in a ball for p-Laplace equation with positive nonlinearity, Adv. Differential Equations, (7) (2002), 877-896. 

[31]

P. Pucci, M. Garcia-Huidobro, R. Manasevich and J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights, Ann. Mat. Pura Appl., 185 (2006), suppl., S205–S243. doi: 10.1007/s10231-004-0143-3.

[32]

A. Sfecci, On the structure of radial solutions for some quasilinear elliptic equations, J. Math. Anal. Appl., 470 (2019), 515-531.  doi: 10.1016/j.jmaa.2018.10.019.

show all references

References:
[1]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.

[2]

A. AmbrosettiJ. Garcia Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242.  doi: 10.1006/jfan.1996.0045.

[3]

B. BarriosJ. Garca-Melin and L. Iturriaga, Semilinear elliptic equations and nonlinearities with zeros, Nonlinear Anal., 134 (2016), 117-126.  doi: 10.1016/j.na.2015.12.025.

[4]

M. F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal., 107 (1989), 293-324.  doi: 10.1007/BF00251552.

[5]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

[6]

F. Dalbono and M. Franca, Nodal solutions for supercritical Laplace equations, Commun in Math. Phys., 347 (2016), 875-901.  doi: 10.1007/s00220-015-2546-y.

[7]

D. G. De FigueiredoJ. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.  doi: 10.1016/S0022-1236(02)00060-5.

[8]

D. G. de FigueiredoP.-L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63. 

[9]

O. Došlý and P. Řehák, Half-linear Differential Equations, North-Holland Mathematics Studies, 202. Elsevier Science B.V., Amsterdam, 2005.

[10]

I. Flores and M. Franca, Phase plane analysis for radial solutions to supercritical quasilinear elliptic equations in a ball, Nonlinear Anal., 125 (2015), 128-149.  doi: 10.1016/j.na.2015.04.015.

[11]

R. H. Fowler, Further studies of Emden's and similar differential equations, Quart. Jl. Math., 2 (1931), 259-288.  doi: 10.1093/qmath/os-2.1.259.

[12]

M. Franca, Classification of positive solution of $p$-Laplace equation with a growth term, Arch. Math. (Brno), 40 (2004), 415-434. 

[13]

M. Franca, A dynamical approach to the study of radial solutions for $p$-Laplace equation, Rend. Semin. Mat. Univ. Politec. Torino, 65 (2007), 53-88. 

[14]

M. Franca, Radial ground states and singular ground states for a spatial dependent $p$-Laplace equation, J. Differential Equations, 248 (2010), 2629-2656.  doi: 10.1016/j.jde.2010.02.012.

[15]

M. Franca, Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type, Ann. Mat. Pura Appl., 189 (2010), 67-94.  doi: 10.1007/s10231-009-0101-1.

[16]

M. Franca and A. Sfecci, Entire solutions of superlinear problems with indefinite weights and Hardy potentials, J. Dyn. Differential Equations, 30 (2018), 1081-1118.  doi: 10.1007/s10884-017-9589-z.

[17]

B. FranchiE. Lanconelli and J. Serrin, Existence and uniqueness of nonnegative solutions of quasilinear equations in ${\mathbb{R}}^n$, Adv. in Math., 118 (1996), 177-243.  doi: 10.1006/aima.1996.0021.

[18]

J. Garca-Melin and L. Iturriaga, Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros, Israel J. Math., 210 (2015), 233-344.  doi: 10.1007/s11856-015-1251-z.

[19]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astron. and Astroph., 24 (1973), 229-238. 

[20]

L. IturriagaS. Lorca and E. Massa, Positive solutions for the $p$-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. H. Poincare Anal. Non Linaire, 27 (2010), 763-771.  doi: 10.1016/j.anihpc.2009.11.003.

[21]

L. IturriagaS. Lorca and E. Massa, Multiple positive solutions for the m-Laplacian and a nonlinearity with many zeros, Differential Integral Equations, 30 (2017), 145-159. 

[22]

L. IturriagaE. MassaJ. Snchez and P. Ubilla, Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327.  doi: 10.1016/j.jde.2009.08.008.

[23]

R. JohnsonX. B. Pan and Y. F. Yi, Singular ground states of semilinear elliptic equations via invariant manifold theory, Nonlinear Anal., 20 (1993), 1279-1302.  doi: 10.1016/0362-546X(93)90132-C.

[24]

R. JohnsonX. B. Pan and Y. F. Yi, The Melnikov method and elliptic equation with critical exponent, Indiana Univ. Math. J., 43 (1994), 1045-1077.  doi: 10.1512/iumj.1994.43.43046.

[25]

N. KawanoW. M. Ni and S. Yotsutani, A generalized Pohozaev identity and its applications, J. Math. Soc. Japan, 42 (1990), 541-564.  doi: 10.2969/jmsj/04230541.

[26]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.  doi: 10.1137/1024101.

[27]

Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398.  doi: 10.1006/jfan.1999.3446.

[28]

W. M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982), 801-807.  doi: 10.1512/iumj.1982.31.31056.

[29]

S. I. Pohozaev, Eigenfunctions of the equations $ \Delta u+ \lambda f(u) = 0$, Soviet Math. Dokl., 165 (1965), 1408-1411. 

[30]

S. Prashanth and K. Sreenadh, Multiplicity results in a ball for p-Laplace equation with positive nonlinearity, Adv. Differential Equations, (7) (2002), 877-896. 

[31]

P. Pucci, M. Garcia-Huidobro, R. Manasevich and J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights, Ann. Mat. Pura Appl., 185 (2006), suppl., S205–S243. doi: 10.1007/s10231-004-0143-3.

[32]

A. Sfecci, On the structure of radial solutions for some quasilinear elliptic equations, J. Math. Anal. Appl., 470 (2019), 515-531.  doi: 10.1016/j.jmaa.2018.10.019.

Figure 3.  Sketch of the graph of the function $ R(d) $. In both the graph we assume $ p< l_s < p^* $ and $ q_s >2 $: we are in the setting of Proposition 4 on the left, picture (a), and in the setting of Proposition 1 on the right, picture (b). In picture (a) we have drawn with solid line the part of the graph constructed using the original problem, and with the dotted line the part constructed using a modified problem to which Proposition 1 could be applied. In both graphs (a) and (b) we have at least $ 3 $ values $ d_i $, such that $ R(d_i) = R $ when $ R>R_2 $; further $ 0<d_1<d_2<U<d_3 $. Moreover in picture $ (a) $ we have $ d_1 \to 0^+ $, $ d_2 \to U^- $ and $ d_3 \to U^+ $ as $ R \to +\infty $, while in picture $ (b) $ we have $ d_1 \to 0^+ $, $ d_2 \to U^- $ and $ d_3 \to M^+ $ as $ R \to +\infty $
Figure 4.  Sketch of the graph of the function $ R(d) $ in the setting of Proposition 6 on the left, and in the setting of Proposition 2 on the right
Figure 5.  Sketch of the graph of the function $ R(d) $ in the setting of Proposition 3 on the left, and in the setting of Proposition 5 on the right. We have drawn with solid lines the part of the graph which has been constructed through the Propositions and with dotted lines the part of the graph constructed through modified problem (for $ d $ large in fig. (a)) or which are just conjectured (for $ d $ small in both fig. (a) and (b)). In both graphs (a) and (b) we have at least $ 2 $ values $ d_i $, such that $ R(d_i) = R $ when $ R>R_2 $; further $ 0< d_2<U<d_3 $. Moreover in picture $ (a) $ we have $ d_2 \to U^- $ and $ d_3 \to U^+ $ as $ R \to +\infty $, while in picture $ b) $ we have $ d_2 \to U^- $ and $ d_3 \to M^+ $ as $ R \to +\infty $
Figure 1.  Sketch of the phase portrait of the autonomous system (12) where $ g_l(x, t)\equiv x|x|^{q-2} $, $ q>2 $, when $ p<l \le p_* $ on the left (a), and when $ p_*<l<p^* $ on the right (b). The manifold $ M^u $ is the solid (black) curve; in fig. (a) the dotted (magenta) curve denotes a trajectory $ \chi(t) $ converging to the origin as $ t \to -\infty $ but not staying in the strongly unstable manifold $ M^u $, in fig (b) the dashed (blue) curve denotes the stable manifold
Figure 2.  Sketch of the phase portrait of the autonomous system (12) where $ g_l(x,t)\equiv c x|x|^{q-2} $, $ q>2 $, when $ l = p^* $ on the left (a), and when $ l>p^* $ on the right (b). In fig. (a) the stable and the unstable manifold $ M^u $ and $ M^s $ coincide. The red line indicates a periodic trajectory corresponding to singular solutions. In fig. (b) the (black) solid line indicates $M^u$ and the dotted (blue) line denotes the stable manifold $ M^s $. They are respectively inside and outside the curve enclosed by the level set $ H=0 $ (green line), due to the Pohozaev identity.
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