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Multiple solutions for nonhomogeneous elliptic equations involving critical Sobolev exponent*

  • Dao-Min Cao (a1), Gong-Bao Li (a1) and Huan-Song Zhou (a1)

Abstract

We consider the following problem:

where is continuous on RN and h(x)≢0. By using Ekeland's variational principle and the Mountain Pass Theorem without (PS) conditions, through a careful inspection of the energy balance for the approximated solutions, we show that the probelm (*) has at least two solutions for some λ* > 0 and λ ∈ (0, λ*). In particular, if p = 2, in a different way we prove that problem (*) with λ ≡ 1 and h(x) ≧ 0 has at least two positive solutions as

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1Ambrosetti, A. and Rabinowitz, P. H.. Dual variational methods in critical point theory and applications. J. Fund. Anal. 14 (1973), 349381.
2Brezis, H. and Lieb, E.. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), 480490.
3Brezis, H. and Nirenberg, L.. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437477.
4Cao, D. M., Li, G. B. and Zhou, H. S.. The existence of two solutions to quasilinear elliptic equations on RN (Preprint).
5Cao, D. M. and Zhou, H. S.. Multiple positive solutions of nonhomogeneous semilinear elliptic equations in RN (Preprint).
6Deng, Y. B.. Existence of multiple positive solutions for - Δu + c2u = u(N + 2)/(N−2) + νf(x) in RN. Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), 161175.
7Graham-Eagle, J.. Monotone methods for semilinear elliptic equations in unbounded domains. J. Math. Anal. Appl. 137 (1989), 122131.
8Ekeland, I.. Nonconvex minimization problems. Bull. Amer. Math. Soc. 3 (1979), 443474.
9Li, G. B.. The existence of a weak solution of quasilinear elliptic equations with critical Sobolev exponent on unbounded domains. Acta Math. Sri. 14 (1994), 6474.
10Li, G. B. and Zhou, H. S.. The existence of a weak solutions of inhomogeneous quasilinear elliptic equations with critical growth conditions. To appear in Acta Math. Sinica, New Series.
11Talenti, G.. Best constant in Sobolev inequality. Ann. Mat. Pure Appl. 110 (1976), 353372.
12Tarantello, G.. On nonhomogeneous elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincare Anal. Non Lineaire 9 (1992), 243261.
13Zhu, X. P.. Nontrivial solution of quasilinear elliptic equations involving critical Sobolev exponents. Scientia Sinica A 31 (1988), 11611181.

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