Hénon type equations with one-sided exponential growth
Keywords
Hénon type equation, Ambrosetti-Prodi problem, critical growthAbstract
We deal with the following class of problems: \begin{equation*} \begin{cases} -\Delta u=\lambda u+|x|^{\alpha}g(u_+)+ f(x)&\mbox{in } B_1,\\ u =0&\mbox{on }\partial B_1, \end{cases} \end{equation*} where $B_1$ is the unit ball in $\mathbb R^2$, $g$ is a $C^1$-function in $[0,+\infty)$ which is assumed to be in the subcritical or critical growth range of Trudinger-Moser type and $f\in L^{\mu}(B_1)$ for some $\mu> 2$. Under suitable hypotheses on the constant $\lambda$, we prove existence of at least two solutions to this problem using variational methods. In case of $f$ radially symmetric, the two solutions are radially symmetric as well.References
A. Ambrosetti, and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1972), 231–247.
M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud. 4 (2004), 453–467.
D. Bonheure, E. Serra and M. Tarallo, Symmetry of extremal functions in Moser–Trudinger inequalities and a Hénon type problem in dimension two, Adv. Differential Equations 13 (2008), 105–138.
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 54 (1991), 939–963.
M. Calanchi and B. Ruf, Elliptic equations with one-sided critical growth, Elec. J. Differential Equations 2002 (2002), 1–21.
M. Calanchi, B. Ruf and Z. Zhang, Elliptic equations in R2 with one-sided exponential growth, Commun. Comtemp. Math. 6 (2004), 947–971.
M. Calanchi and E. Terraneo, Non-radial maximizers for functionals with exponential non-linearity in R2 , Adv. Nonlinear Stud. 5 (2005), 337–350.
D. de Figueiredo, Lectures on Boundary Value Problems of Ambrosetti–Prodi Type, Atas do 12o Seminario Brasileiro de Análise, São Paulo, 1980.
D. de Figueiredo, The Ekeland Variational Principle with Applications and Detours, Tata Inst. Fund. Res. Lectures on Math. and Phys., vol. 81, Springer, Berlin, 1989.
D. de Figueiredo, O. Miyagaki and B. Ruf, Elliptic equations R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), 139–153.
D. de Figueiredo and J. Yang, Critical superlinear Ambrosetti–Prodi problems, Topol. Methods Nonlinear Anal. 14 (1999), 59–80.
F. Gazzola and B. Ruf, Lower order pertubations of critical growth nonlinearities in semilinear elliptic equations, Adv. Differential Equations 2 (1997), 555–572.
M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronomy and Astrophysics 24 (1973), 229–238.
J. Kazdan and F. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567–597.
W. Long and J. Yang, Existence for critical Hénon-type equations, Differential Intntegral Equations 25 (2012), 567–578.
J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.
J. Moser, A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J. 20 (1971), 1077–1092.
W. Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Ind. Univ. Math. J. 31 (1982), 801–807.
R. Palais, The principle of symmetric criticality, Commum. Math Phys. 69 (1979), 19–30.
P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1986.
M. Ramos, Teoremas de Multiplicidade do Tipo Ambrosetti–Prodi, Textos e Notas, CMAF, Lisboa–Portugal, 1988.
B. Ribeiro, Critical elliptic problems in R2 involving resonance in high-order eigenvalues, Commun. Contemp. Math. 17 (2015), 22 pp.
B. Ribeiro, The Ambroseti–Prodi problem for elliptic systems with Trudinger–Moser nonlinearity, Proc. Edinburgh Math. Soc. 55 (2012), 215–244.
S. Secchi, E. Serra, Symmetry breaking results for problems with exponential growth in the unit disk, Commun. Contemp. Math. 8 (2006), 823–839.
E. Serra, Nonradial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations 23 (2005), 301–326.
D. Smets, J. Su, M. Willem, Non-radial ground states for the Hénon equation, Commun. Contemp. Math. 4 (2002), 467–480.
N. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483.
D. Yinbin, On the superlinear Ambrosetti–Prodi problem involving critical Sobolev exponents, Nonlinear Anal. 17 (1991), 1111–1124.
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