## Existence of positive solutions of the equation $$-\Delta u+a(x)u=u^{(N+2)/(N-2)}$$ in $${\mathbb{R}}^ N$$.(English)Zbl 0705.35042

Consider the following problem: find u such that $(*)\;u(x)>0,\;-\Delta u+a(x)u=| u|^{2^*-2}u\text{ on } \Omega,\;u\in {\mathcal D}^{1,2}(\Omega),$ where $$\Omega \subseteq {\mathbb{R}}^ N$$, $$N\geq 3$$, $$2^*=2N/(n-2)$$, $$a(x)\geq 0$$ and $${\mathcal D}^{1,2}(\Omega)$$ is a closure of $$C^{\infty}_ 0(\Omega)$$ with respect to the standard Sobolev norm of $$H^ 1(\Omega)$$. Then, they prove the following Theorem. If $$\Omega ={\mathbb{R}}^ N$$ and $$a(x)\geq 0\forall x\in {\mathbb{R}}^ N$$ and $$a(x)\geq \nu >0$$ in a neighborhood of a point $$\bar x,$$ $$\exists p_ 1<N/2$$ and $$p_ 2>N/2$$ and for $$N=3$$, $$p_ 2<3$$, such that $$a(x)\in L^ p\forall p\in [p_ 1,p_ 2]$$, $$| a|_{L^{N/2}}<S(2^{2/N}-1)$$, where $S=\inf \{\int_{{\mathbb{R}}^ N}| \nabla u|^ 2 dx/(\int_{{\mathbb{R}}^ N}| u|^{2^*} dx)^{2/2^*};\;u\in {\mathcal D}^{1,2}({\mathbb{R}}^ N)\},$ then the problem (*) has at least one positive solution.
Reviewer: Y.Ebihara

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

critical value; positive solution
Full Text:

### References:

 [1] Aubin, Th., Problèmes isopérimétriques et espaces de Sobolev, J. differential geom., 11, 573-598, (1976) · Zbl 0371.46011 [2] Bahri, A.; Coron, J.M., Sur une équation elliptique nonlinéaire avec l’exposant critique de Sobolev, C.R. acad. sci. Paris, 301, 345-348, (1985), (and detailed paper to appear) · Zbl 0601.35040 [3] Benci, V.; Cerami, G., Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. rational mech. anal., 99, 283-300, (1987) · Zbl 0635.35036 [4] Benci, V.; Fortunato, D., Some compact embedding theorems for weighted Sobolev spaces, Boll. un. mat. ital. B (5), 13, 832-843, (1976) · Zbl 0382.46018 [5] Berestycki, H.; Lions, P.L.; Berestycki, H.; Lions, P.L., Nonlinear scalar field equations. (II)-existence of infinitely many solutions, Arch. rational mech. anal., Arch. rational mech. anal., 82, 347-376, (1983) · Zbl 0556.35046 [6] Brezis, H., Elliptic equations with limiting Sobolev exponents—the impact of topology, () · Zbl 0601.35043 [7] Brezis, H.; Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, (), 486-490 · Zbl 0526.46037 [8] Cerami, G.; Solimini, S.; Struwe, M., Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. funct. anal., 69, 289-306, (1986) · Zbl 0614.35035 [9] Coron, J.M., Topologie et cas limite des injections de Sobolev, C.R. acad. sci. Paris Sér. I, 299, 209-212, (1984) · Zbl 0569.35032 [10] {\scW. Ding}, On a conformally invariant equation on $$R$$^{N}, preprint. [11] Gidas, B., Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, (), 255-273 [12] Gidas, B.; Ni, W.M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in $$R$$^{N}, (), 370-401, Part A [13] Hofer, H., Variational and topological methods in partially ordered Hilbert spaces, Math. ann., 261, 493-514, (1982) · Zbl 0488.47034 [14] Lions, P.L.; Lions, P.L., The concentration-compactness principle in the calculus of variations: the limit case, Rev. mat. iberoamericana, Rev. mat. iberoamericana, 1, 145-201, (1985) · Zbl 0704.49005 [15] Pohozaev, S.I., Eigenfunctions of the equation $$Δu + λƒ(u) = 0$$, Sov. math. dokl., 6, 1408-1411, (1965) · Zbl 0141.30202 [16] {\scP. H. Rabinowitz}, Minimax methods in critical point theory with applications to differential equations, in “Amer. Math. Soc. Regional Conference Series in Mathematics,” No. 65. · Zbl 0609.58002 [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. pura appl., 110, 353-372, (1976) · Zbl 0353.46018
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.