zbMATH — the first resource for mathematics

On critical exponents for the Pucci’s extremal operators. (English) Zbl 1274.35115
Summary: In this article we study some results on the existence of radially symmetric non-negative solutions for the nonlinear elliptic equation
\[ \mathcal{M}_{\lambda,\Lambda}^+(D^2u)+u^p=0\quad\text{ in }\;\mathbb{R}^N.\qquad\qquad\qquad\qquad(*) \]
Here \(N\geq 3\), \(p>1\) and \(\mathcal{M}_{\lambda,\Lambda}^+\) denotes the Pucci’s extremal operators with parameters \(0<\lambda\leq\Lambda\). The goal is to describe the solution set in function of the parameter \(p\). We find critical exponents \(1<p_*^+<p_+^p\) that satisfy: (i) If \(1<p<p_+^*\) then there is no non-trivial radial solution of \((*)\). (ii) If \(p=p_+^*\) then there is a unique fast decaying radial solution of \((*)\). (iii) If \(p_+^*<p\leq p_+^p\) then there is a unique pseudo-slow decaying radial solution to \((*)\). (iv) If \(p_+^p<p\) then there is a unique slow decaying radial solution to \((*)\). Similar results are obtained for the operator \(\mathcal{M}_{\lambda,\Lambda}^-\).

35J60 Nonlinear elliptic equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
Full Text: DOI Numdam EuDML
[1] Cabré, X.; Caffarelli, L.A., Fully nonlinear elliptic equation, Colloquium publication, 43, (1995), American Mathematical Society
[2] Caffarelli, L.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. pure appl. math., 42, 3, 271-297, (1989) · Zbl 0702.35085
[3] Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke math. J., 3, 3, 615-622, (1991) · Zbl 0768.35025
[4] Clemons, C.; Jones, C., A geometric proof of kwong – mc leod uniqueness result, SIAM J. math. anal., 24, 436-443, (1993) · Zbl 0779.35040
[5] Coffman, C., Uniqueness of the ground state solution for δu−u+u3=0 and a variational characterization of other solutions, Arch. rational mech. anal., 46, 81-95, (1972) · Zbl 0249.35029
[6] Cutri, A.; Leoni, F., On the Liouville property for fully nonlinear equations, Ann. inst. H. Poincaré analyse non lineaire, 17, 2, 219-245, (2000) · Zbl 0956.35035
[7] Deng, Y.; Cao, D., Uniqueness of the positive solution for singular non-linear boundary value problems, Syst. sci math. sci., 6, 25-31, (1993) · Zbl 0789.34025
[8] Erbe, L.; Tang, M., Structure of positive radial solutions of semilinear elliptic equation, J. differential equations, 133, 179-202, (1997) · Zbl 0871.34023
[9] Gidas, B., Symmetry and isolated singularitiesof positive solutions of nonlinear elliptic equations, (), 255-273
[10] Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. pure appl. math., 34, 525-598, (1981) · Zbl 0465.35003
[11] Hale, J., Ordinary differential equation, (1969), Wiley New York · Zbl 0186.40901
[12] Kajikiya, R., Existence and asymptotic behavior of nodal solution for semilinear elliptic equation, J. differential equations, 106, 238-256, (1993) · Zbl 0791.35039
[13] Kolodner, I., The heavy rotating string – a nonlinear eigenvalue problem, Comm. pure appl. math., 8, 395-408, (1955) · Zbl 0065.17202
[14] Kwong, M.K., Uniqueness of positive solution of δu−u+up=0 in \(R\^{}\{N\}\), Arch. rational mech. anal., 105, 243-266, (1989) · Zbl 0676.35032
[15] Kwong, M.K.; Zhang, L., Uniqueness of positive solution of δu+f(u)=0 in an annulus, Differential integral equations, 4, 583-596, (1991) · Zbl 0724.34023
[16] Ni, W.M.; Nussbaum, R., Uniqueness and nonuniqueness for positive radial solutions of δu+f(u,r)=0, Comm. pure appl. math., 38, 67-108, (1985) · Zbl 0581.35021
[17] Pohozaev, S.I., Eigenfunctions of the equation δu+λf(u)=0, Soviet math., 5, 1408-1411, (1965) · Zbl 0141.30202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.