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doi: 10.3934/dcds.2021037

Supercritical elliptic problems involving a Cordes like operator

Craig Cowan

University of Manitoba, Winnipeg, MB, R3T 2N2, Canada

Received  August 2019 Revised  November 2020 Published  March 2021

Fund Project: The first author is supported by NSERC Discovery Grant

In this work we obtain positive bounded solutions of various perturbations of
$ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u - \gamma \sum_{i, j = 1}^N \frac{x_i x_j}{|x|^2} u_{x_i x_j} & = & u^p \qquad \mbox{ in } B_1, \\ \hfill u & = & 0 \hfill\qquad\ \mbox{ on } \partial B_1, \end{array}\right. \end{equation} \ \ \ \ \ \ \ \ \ \ \ (1) $
where
$ B_1 $
 is the unit ball in
$ {{\mathbb{R}}}^N $
 where
$ N \ge 3 $
,
$ \gamma>0 $
 and
$ 1<p<p_{N, \gamma} $
 where
$ \begin{equation*} p_{N, \gamma}: = \left\{ \begin{array}{lc} \frac{N+2+3 \gamma}{N-2-\gamma} & \qquad \mbox{ if } \gamma<N-2, \\ \infty & \qquad \mbox{ if } \gamma \ge N-2. \end{array}\right. \end{equation*} $
Note for
$ \gamma>0 $
 this allows for supercritical range of
$ p $
.
Keywords: Elliptic pde, supercritical, Cordes operator.
Mathematics Subject Classification: Primary: 35J15; Secondary: 35J61.
Citation: Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021037
References:
[1]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[2]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[3]

M. Chicco, Equazioni ellittiche del secondo ordine di tipo Cordes con termini di ordine inferiore, Ann. Mat. Pura Appl., 85 (1970), 347-356.  doi: 10.1007/BF02413544.  Google Scholar

[4]

M. Clapp, M. Grossi and A. Pistoia, Multiple solutions to the Bahri-Coron problem in domains with a shrinking hole of positive dimension, Complex Var. and Elliptic Eqns., 57 (2012), 1147-1162.  doi: 10.1080/17476931003628265.  Google Scholar

[5]

H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, Proc. Symp. Pure Math., 4 (1961), 157-166.   Google Scholar

[6]

J. M. Coron, Topologie et cas limite des injections de Sobolev, C.R. Acad. Sc. Paris, Series I, 299 (1984), 209-212.   Google Scholar

[7]

L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 631-652.  doi: 10.1016/S0294-1449(99)80030-4.  Google Scholar

[8]

J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680.  doi: 10.1142/S0219199707002575.  Google Scholar

[9]

M. del Pino, Supercritical elliptic problems from a perturbation viewpoint, Discrete and Continuous Dynamical Systems, 21 (2008), 69-89.  doi: 10.3934/dcds.2008.21.69.  Google Scholar

[10]

M. del Pino, P. Felmer and Mo nica Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calculus of Variations and Partial Differential Equations, 16 (2003), 113-145.  doi: 10.1007/s005260100142.  Google Scholar

[11]

M. del Pino, P. Felmer and M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries, Bull. London Math. Society, 35 (2003), 513-521.  doi: 10.1112/S0024609303001942.  Google Scholar

[12]

M. del Pino and M. Musso, Super-critical bubbling in elliptic boundary value problems, Variational Problems and Related Topics (Kyoto, 2002), 1307 (2003), 85-108.   Google Scholar

[13]

G. Di Fazio, D. I. Hakim and Y. Sawano, Elliptic equations with discontinuous coefficients in generalized Morrey spaces, European Journal of Mathematics, 3 (2017), 728-762.  doi: 10.1007/s40879-017-0168-y.  Google Scholar

[14]

B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[15]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[16]

F. Gladiali and M. Grossi, Supercritical elliptic problem with nonautonomous nonlinearities, J. Diff. Eqns., 253 (2012), 2616-2645.  doi: 10.1016/j.jde.2012.07.006.  Google Scholar

[17]

M. Grossi and F. Takahashi, Nonexistence of multi-bubble solutions to some elliptic equations on convex domains, Jour. Funct. Anal., 259 (2010), 904-917.  doi: 10.1016/j.jfa.2010.03.008.  Google Scholar

[18]

M. Hieber and I. Wood, The Dirichlet problem in convex bounded domains for operators in non-divergence form with $L^\infty$ coefficients, Differential and Integral Equations, 20 (2007), 721-734.   Google Scholar

[19]

P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific Publishing Co. Pte. Ltd.2012. 256 pp. doi: 10.1142/8308.  Google Scholar

[20]

C. S. Lin and W. M. NI, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Mat. Soc., 102 (1988), 271-277.  doi: 10.1090/S0002-9939-1988-0920985-9.  Google Scholar

[21]

A. Maugeri, D. K. Palagachev and L. G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, Wiley, Berlin, 2000. doi: 10.1002/3527600868.  Google Scholar

[22]

R. Mazzeo and F. Pacard, A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Diff. Geom., 44 (1996), 331-370.  doi: 10.4310/jdg/1214458975.  Google Scholar

[23]

D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal., 114 (1993), 97-105.  doi: 10.1006/jfan.1993.1064.  Google Scholar

[24]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl., 6 (1965), 1408-1411.   Google Scholar

[25]

M. Struwe, Variational Methods–Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Berlin: Springer-Verlag, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar

[26]

G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.  doi: 10.1007/BF02414375.  Google Scholar

[27]

G. Talenti, Equazioni lineari ellittiche in due variabili, Matematiche, 21 (1966), 339-376.   Google Scholar

[28]

K. Wang and J. Wei, Analysis of blow-up locus and existence of weak solutions for nonlinear supercritical problems, International Mathematics Research Notices, 2015 (2015), 2634-2670.  doi: 10.1093/imrn/rnu013.  Google Scholar

show all references

References:
[1]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.  Google Scholar

[2]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[3]

M. Chicco, Equazioni ellittiche del secondo ordine di tipo Cordes con termini di ordine inferiore, Ann. Mat. Pura Appl., 85 (1970), 347-356.  doi: 10.1007/BF02413544.  Google Scholar

[4]

M. Clapp, M. Grossi and A. Pistoia, Multiple solutions to the Bahri-Coron problem in domains with a shrinking hole of positive dimension, Complex Var. and Elliptic Eqns., 57 (2012), 1147-1162.  doi: 10.1080/17476931003628265.  Google Scholar

[5]

H. O. Cordes, Zero order a priori estimates for solutions of elliptic differential equations, Proc. Symp. Pure Math., 4 (1961), 157-166.   Google Scholar

[6]

J. M. Coron, Topologie et cas limite des injections de Sobolev, C.R. Acad. Sc. Paris, Series I, 299 (1984), 209-212.   Google Scholar

[7]

L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), 631-652.  doi: 10.1016/S0294-1449(99)80030-4.  Google Scholar

[8]

J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680.  doi: 10.1142/S0219199707002575.  Google Scholar

[9]

M. del Pino, Supercritical elliptic problems from a perturbation viewpoint, Discrete and Continuous Dynamical Systems, 21 (2008), 69-89.  doi: 10.3934/dcds.2008.21.69.  Google Scholar

[10]

M. del Pino, P. Felmer and Mo nica Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calculus of Variations and Partial Differential Equations, 16 (2003), 113-145.  doi: 10.1007/s005260100142.  Google Scholar

[11]

M. del Pino, P. Felmer and M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries, Bull. London Math. Society, 35 (2003), 513-521.  doi: 10.1112/S0024609303001942.  Google Scholar

[12]

M. del Pino and M. Musso, Super-critical bubbling in elliptic boundary value problems, Variational Problems and Related Topics (Kyoto, 2002), 1307 (2003), 85-108.   Google Scholar

[13]

G. Di Fazio, D. I. Hakim and Y. Sawano, Elliptic equations with discontinuous coefficients in generalized Morrey spaces, European Journal of Mathematics, 3 (2017), 728-762.  doi: 10.1007/s40879-017-0168-y.  Google Scholar

[14]

B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.  Google Scholar

[15]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[16]

F. Gladiali and M. Grossi, Supercritical elliptic problem with nonautonomous nonlinearities, J. Diff. Eqns., 253 (2012), 2616-2645.  doi: 10.1016/j.jde.2012.07.006.  Google Scholar

[17]

M. Grossi and F. Takahashi, Nonexistence of multi-bubble solutions to some elliptic equations on convex domains, Jour. Funct. Anal., 259 (2010), 904-917.  doi: 10.1016/j.jfa.2010.03.008.  Google Scholar

[18]

M. Hieber and I. Wood, The Dirichlet problem in convex bounded domains for operators in non-divergence form with $L^\infty$ coefficients, Differential and Integral Equations, 20 (2007), 721-734.   Google Scholar

[19]

P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific Publishing Co. Pte. Ltd.2012. 256 pp. doi: 10.1142/8308.  Google Scholar

[20]

C. S. Lin and W. M. NI, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Mat. Soc., 102 (1988), 271-277.  doi: 10.1090/S0002-9939-1988-0920985-9.  Google Scholar

[21]

A. Maugeri, D. K. Palagachev and L. G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, Wiley, Berlin, 2000. doi: 10.1002/3527600868.  Google Scholar

[22]

R. Mazzeo and F. Pacard, A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Diff. Geom., 44 (1996), 331-370.  doi: 10.4310/jdg/1214458975.  Google Scholar

[23]

D. Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal., 114 (1993), 97-105.  doi: 10.1006/jfan.1993.1064.  Google Scholar

[24]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl., 6 (1965), 1408-1411.   Google Scholar

[25]

M. Struwe, Variational Methods–Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Berlin: Springer-Verlag, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar

[26]

G. Talenti, Sopra una classe di equazioni ellittiche a coefficienti misurabili, Ann. Mat. Pura Appl., 69 (1965), 285-304.  doi: 10.1007/BF02414375.  Google Scholar

[27]

G. Talenti, Equazioni lineari ellittiche in due variabili, Matematiche, 21 (1966), 339-376.   Google Scholar

[28]

K. Wang and J. Wei, Analysis of blow-up locus and existence of weak solutions for nonlinear supercritical problems, International Mathematics Research Notices, 2015 (2015), 2634-2670.  doi: 10.1093/imrn/rnu013.  Google Scholar

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