## An existence result for nonliner elliptic problems involving critical Sobolev exponent.(English)Zbl 0612.35053

The authors study the problem $-\Delta u-\lambda u=| u|^{2^*-2}u\quad on\quad \Omega;\quad u=0\quad on\quad \partial \Omega,$ where $$\Omega$$ is a bounded domain in $${\mathbb{R}}^ n$$, $$\lambda$$ is a real parameter, $$2^*=2n/(n-2)$$ is the critical Sobolev exponent for the embedding $$H^ 1_ 0(\Omega)\subset L_ p(\Omega)$$. It is proved that in the case $$n\geq 4$$, for any $$\lambda\geq 0$$ there exists at least one nontrivial solution $$u\in H^ 1_ 0(\Omega)$$.
Reviewer: M.Kučera

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35J20 Variational methods for second-order elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000)

### Keywords:

existence; critical Sobolev exponent; embedding
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### References:

 [1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical points theory and applications, J. Funct. Analysis, t. 14, 349-381, (1973) · Zbl 0273.49063 [2] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with « strong resonance » at infinity, Journal of nonlinear Anal. T. M. A., t. 7, 981-1012, (1983) · Zbl 0522.58012 [3] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., t. XXXVI, 437-477, (1983) · Zbl 0541.35029 [4] A. Capozzi, G. Palmieri, Multiplicity results for nonlinear elliptic equations involving critical Soboler exponent, preprint. · Zbl 0624.35035 [5] Cerami, G.; Fortunato, D.; Struwe, M., Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré. Analyse non linéaire, t. 1, 341-350, (1984) · Zbl 0568.35039 [6] D. Fortunato, Problemi ellittici con termine non lineare a crescita critica, Proceedings of the meeting « Problemi differenziali e teoria dei punti critici ». Bari, marzo, 1984. [7] Pohozaev, S. J., Eigenfunctions of the equation δu + λf(u) = 0, Soviet Math. Doklady, t. 6, 1408-1411, (1965) · Zbl 0141.30202 [8] Struwe, M., A global compactness result for elliptic boundary value problems involving limiting non-linearities, Math. Z., t. 187, 511-517, (1984) · Zbl 0535.35025
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