## The role played by space dimension in elliptic critical problems.(English)Zbl 0938.35058

Weak $$H_{0}^{1}(\Omega)$$ solutions of the nonlinear critical second order elliptic problem $\begin{cases}-\Delta u-\mu u/|x|^{2}=u^{2^{\ast }-1}+\lambda u & \text{in $$\Omega$$},\\ u>0 & \text{in }\Omega,\\ u=0 & \text{on }\partial \Omega\end{cases}$ are considered, where $$\Omega$$ is a smooth bounded domain in $$\mathbb{R}^{N}$$, $$N\geq 3$$, such that $$0\in \Omega$$. Here $$2^{\ast }=2N/(N-2)$$ is the so-called critical exponent, $$\mu$$ is a real parameter and $$\lambda >0.$$ One can say that a dimension $$N$$ is critical for the problem (1) if there exists a smooth domain in $$\mathbb{R}^{N}$$ in which (1) has no solutions for some $$\lambda \in (0,\lambda _{1}),$$ where $$\lambda _{1}$$ is the first eigenvalue of $$-\Delta$$ in $$\Omega$$ [see for instance P. Pucci and J. Serrin, J. Math. Pures Appl. 69, 55-83 (1990; Zbl 0717.35032)]. In H. Brezis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437-477 (1983; Zbl 0541.35029)], was already proved that the dimension $$N\geq 4$$ is not critical, and $$N=3$$ is critical for (1) with $$\mu =0.$$ In the paper, these results are generalized for the case $$\mu \neq 0.$$ Moreover, it is shown that any fixed dimension $$N\geq 3$$ may be critical or not, as follows: if $$\mu \leq \overline{\mu }-1$$ then $$N$$ is not critical and if $$\overline{\mu }-1<\mu <\overline{\mu }$$ then $$N$$ is critical, where $$\overline{\mu }=(N-2)^{2}/4.$$

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations

### Citations:

Zbl 0717.35032; Zbl 0541.35029
Full Text:

### References:

  Bernis, F.; Grunau, H.-Ch., Critical exponents and multiple critical dimensions for polyharmonic operators, J. differential equations, 117, 469-486, (1995) · Zbl 0831.35057  Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical exponents, Comm. pure appl. math., 36, 437-477, (1983) · Zbl 0541.35029  H. Brezis, and, J. L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, in preparation. · Zbl 0894.35038  Edmunds, D.E.; Fortunato, D.; Jannelli, E., Critical exponents, critical dimensions and the biharmonic operator, Arch. rational mech. anal., 112, 269-289, (1990) · Zbl 0724.35044  Gazzola, F., Critical growth problems for polyharmonic operators, Proc. roy. soc. Edinburgh sect. A, 128, 251-263, (1998) · Zbl 0926.35034  Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020  Azorero, J.Garcia; Alonso, I.Peral, Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues, Comm. partial differential equations, 12, 1389-1430, (1987) · Zbl 0637.35069  Grunau, H.-Ch., Critical exponents and multiple critical dimensions for polyharmonic operators, Boll. un. mat. ital. B (7), II, 815-847, (1995) · Zbl 0854.35040  Grunau, H.-Ch., On a conjecture of P. pucci and J. Serrin, Analysis, 16, 399-403, (1996) · Zbl 0862.35033  Hardy, G.; Littlewood, J.E.; Polya, G., Inequalities, (1934), Cambridge Univ. Press Cambridge · JFM 60.0169.01  E. Jannelli, and, S. Solimini, Critical behaviour of some elliptic equations with singular coefficients, in preparation. · Zbl 0944.35022  Noussair, E.S.; Swanson, Ch.A.; Jianfu, Yang, Critical semilinear biharmonic equations in $$R$$^{N}, Proc. roy. soc. Edinburgh sect. A, 121, 139-148, (1992) · Zbl 0779.35044  Pohozaev, S., Eigenfunction of the equation δu+λf(u)=0, Soviet math. dokl., 6, 1408-1411, (1965) · Zbl 0141.30202  Pucci, P.; Serrin, J., A general variational identity, Indiana univ. math. J., 35, 681-703, (1986) · Zbl 0625.35027  Pucci, P.; Serrin, J., Critical exponents and critical dimensions for polyharmonic operators, J. math. pures appl., 69, 55-83, (1990) · Zbl 0717.35032
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.