×

A note on a critical problem with natural growth in the gradient. (English) Zbl 1245.35032

Summary: The paper analyzes the influence on the meaning of “natural growth in the gradient”, of a perturbation by a Hardy potential in some elliptic equations. We obtain a linear differential operator that, in a natural way, is the corresponding gradient for the perturbed elliptic problem. Indeed, in the case of the Laplacian the natural problem becomes \[ -\Delta u-\Lambda_N\frac u{| x|^2}=| \nabla u+\frac{N-2}2 \frac u{| x|^2}|^2 | x|^{(N-2)/2}+\lambda f(x)\text{ in }\Omega,\quad u=0\text{ on }\partial\Omega, \] \(\Lambda_N=((N-2)/2)^2\).
The main results are: i) Optimal summability of the data to have weak solutions; ii) Optimal linear operator associated, and, iii) Multiplicity and characterization of all solutions in terms of some measures. The results also are new for the Laplace operator perturbed for an inverse-square potential.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35D30 Weak solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abdellaoui, B., Dall’Aglio, A., Peral, I.: Some remarks on elliptic problems with criti- cal growth on the gradient. J. Differential Equations 222 , 21-62 (2006) · Zbl 1357.35089
[2] Abdellaoui, B., Peral, I.: On quasilinear elliptic equations related to some Caffarelli-Kohn- Nirenberg inequalities. Comm. Pure Appl. Anal. 2 , 539-566 (2003) · Zbl 1148.35324 · doi:10.3934/cpaa.2003.2.539
[3] Abdellaoui, B., Colorado, E., Peral, I.: Some improved Caffarelli-Kohn-Nirenberg in- equalities. Calc. Var. Partial Differential Equations 23 , 327-345 (2005) · Zbl 1207.35114 · doi:10.1007/s00526-004-0303-8
[4] Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J. L.: An L1 theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa 22 , 241-273 (1995) · Zbl 0866.35037
[5] Berestycki, H., Kamin, S., Sivashinsky, G.: Metastability in a flame front evolution equation. Interfaces Free Bound. 3 , 361-392 (2001) · Zbl 0991.35097 · doi:10.4171/IFB/45
[6] Boccardo, L., Murat, F., Puel, J.-P.: Existence de solutions non bornées pour cer- taines équations quasi-linéaires. Portugal. Math. 41 , 507-534 (1982) · Zbl 0524.35041
[7] Boccardo, L., Segura de León, S., Trombetti, C.: Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term. J. Math. Pures Appl. 80 , 919-940 (2001) · Zbl 1134.35358 · doi:10.1016/S0021-7824(01)01211-9
[8] Brezis, H., Cabré, X.: Some simple nonlinear PDE’s without solution. Boll. Un. Mat. Ital. Sez. B (8) 1 , 223-262 (1998) · Zbl 0907.35048
[9] Brezis, H., Dupaigne, L., Tesei, A.: On a semilinear equation with inverse-square potential. Selecta Math. 11 , 1-7 (2005) · Zbl 1161.35383 · doi:10.1007/s00029-005-0003-z
[10] Brezis, H., Vázquez, J. L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10 , 443-469 (1997) · Zbl 0894.35038
[11] Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equa- tions with general measure data. Ann. Scuola Norm. Sup. Pisa 28 , 741-808 (1999) · Zbl 0958.35045
[12] Ferone, V., Murat, F.: Nonlinear problems having natural growth in the gradient: an ex- istence result when the source terms are small. Nonlinear Anal. 42 , 1309-1326 (2000) · Zbl 1158.35358 · doi:10.1016/S0362-546X(99)00165-0
[13] Hansson, K., Maz’ya, V. G., Verbitsky, I. E.: Criteria of solvability for multidimensional Ric- cati equations. Ark. Mat. 37 , 87-120 (1999) · Zbl 1087.35513 · doi:10.1007/BF02384829
[14] Kardar, M., Parisi, G., Zhang, Y. C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 , 889-892 (1986) · Zbl 1101.82329 · doi:10.1103/PhysRevLett.56.889
[15] Lions, P. L., Generalized Solutions of Hamilton-Jacobi Equations. Res. Notes Math. 69, Pit- man (1982) · Zbl 0497.35001
[16] Pinchover, Y., Tintarev, K.: Existence of minimizers for Schrödinger equations under domain perturbations with applications to Hardy’s inequality. Indiana Univ. Math. J. 54 , 1061-1074 (2005) · Zbl 1213.35216 · doi:10.1512/iumj.2005.54.2705
[17] Wang, Z. Q., Willem, M.: Caffarelli-Kohn-Nirenberg inequalities with remainder terms. J. Funct. Anal. 203 , 550-568 (2003) · Zbl 1037.26014 · doi:10.1016/S0022-1236(03)00017-X
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.