Some existence results for superlinear elliptic boundary value problems involving critical exponents. (English) Zbl 0614.35035

Cerami, G.; Solimini, S.; Struwe, M.

J. Funct. Anal. 69, 289-306 (1986).

This paper deals with the superlinear elliptic boundary value problem \[ - \Delta u-\lambda u=u | u|^{2^*-2}\quad in\quad \Omega;\quad u|_{\partial \Omega}=0, \] where \(\Omega\) is a smoothly bounded domain in \(R^ n\), \(n>2\), \(\lambda\in R\), \(2^*=2n/(n-2)\) and \(2^*\) is the limiting Sobolev exponent for the embedding \(H^ 1_ 0(\Omega)\to L^ p(\Omega)\). Solving this problem is equivalent to finding critical points in \(H^ 1_ 0(\Omega)\) of the energy functional \[ I_{\lambda}(u)=(1/2)\int_{\Omega}(| \nabla u|^ 2-\lambda | u|^ 2)dx-(1/2)\int_{\Omega}| u|^{2^*} dx. \] First, based on the global compactness theorem for the problem (1), the compactness question is discussed. Then some results about the existence and multiplicity of solutions to the above problem are obtained.

Reviewer: Xu Zhenyuan

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35J20	Variational methods for second-order elliptic equations
35D05	Existence of generalized solutions of PDE (MSC2000)
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

superlinear elliptic boundary value problem; limiting Sobolev exponent; critical points; energy functional; global compactness; existence; multiplicity of solutions

References:

[2]	Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure. Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029
[4]	Cerami, G.; Fortunato, D.; Struwe, M., Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré (Anal. Nonlinéaire), 1, 341-350 (1984) · Zbl 0568.35039
[6]	Gidas, B., Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, (Sternberg, R. L., Nonlinear Differential Equations in Engineering and Applied Sciences (1979), Dekker: Dekker New York), 255-273
[7]	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
[8]	Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), (Nachbin, L., Mathematical Analysis and Applications, Part A (1981), Academic Press: Academic Press Orlando, Fla), 370-401
[9]	Hofer, H., Variational and topological methods in partially ordered Hilbert spaces, Math. Ann., 261, 493-514 (1982) · Zbl 0488.47034
[10]	Loewner, C.; Nirenberg, L., Partial differential equations invariant under conformal and projective transformations, (Contributions to Analysis (1974), Academic Press: Academic Press Orlando, Fla), 245-272
[11]	Miranda, C., Un’osservazione sul teorema di Brouwer, Boll. Un. Mat. Ital. Ser. II, Anno III n. 1, 19, 5-7 (1940) · JFM 66.0217.01
[12]	Palais, R. S., Morse theory on Hilbert manifolds, Topology, 2, 299-340 (1963) · Zbl 0122.10702
[13]	Pohožaev, S. I., Eigenfunctions of the equation \(Δu + λf (u) = 0\), Soviet Math. Dokl., 6, 1408-1411 (1965) · Zbl 0141.30202
[14]	Rabinowitz, P. H., Variational methods for nonlinear eigenvalue problems, (Prodi, G., Eigenvalues in Nonlinear Problems (1974), CIME), 141-195
[15]	Solimini, S., On the solvability of some elliptic partial differential equations with the linear part at resonance, J. Math. Anal. Appl., 117, 138-152 (1986) · Zbl 0634.35030
[16]	Struwe, M., Superlinear elliptic boundary value problems with rotational symmetry, Arch. Math., 39, 233-240 (1982) · Zbl 0496.35034
[17]	Struwe, M., A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187, 511-517 (1984) · Zbl 0535.35025
[18]	Talenti, G., Best constants in Sobolev inequality, Ann. Mat., 110, 353-372 (1976) · Zbl 0353.46018
[19]	Uhlenbeck, K., Variational problems for Gauge fields, (Yau, S. T., Seminar on Differential Geometry (1982), Princeton Univ. Press: Princeton Univ. Press Princeton, N. J) · Zbl 0562.53059