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Positive solutions of quasilinear elliptic obstacle problems with critical exponents. (English) Zbl 0838.49008

In this paper it has been considered a problem of finding positive solutions for a quasilinear elliptic obstacle problem with a critical exponent: find \(u\in K=\{v\in W_0^{1,p} (\Omega): v(x)\geq \varphi(x)\) a.e. in \(\Omega\}\) such that \[ \int_\Omega |Du|^{p-2} Du\cdot D(v-u) dx\geq \lambda \int_\Omega u^{p^*-1} (v-u) dx \qquad \forall v\in K, \tag{0.1} \] where \(\Omega\) is a bounded domain, \(2\leq p< n\), \(p^*\) a critical exponent and \(\varphi\in C^{1, \beta} (\Omega)\) \((\varphi|_{\partial \Omega}< 0\), \(\varphi^+\neq 0)\). The author has shown if \(\lambda\) is not too big that (0.1) has a minimal positive solution by using the Ekeland’s variational principle and that in some cases the problem (0.1) has at least two positive solutions by using a variant mountain pass theorem.

MSC:

49J40 Variational inequalities
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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[1] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. funct. Analysis, 14, 349-381 (1973) · Zbl 0273.49063
[2] Bahri, A.; Coron, J. M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Communs pure appl. Math, 41, 253-294 (1988) · Zbl 0649.35033
[3] Brezis, H.; Lieb, E. H., A relation between pointwise convergence of functions and convergence of integrals, Proc. Am. math. Soc., 88, 486-490 (1983) · Zbl 0526.46037
[4] Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communs pure appl. Math., 36, 437-477 (1983) · Zbl 0541.35029
[5] Coron, J. M., Topologie et cas limite des injections de Sobolev, C.r. Acad. Sci. Paris, 299, 209-212 (1984) · Zbl 0569.35032
[6] Ekeland, I., Nonconvex minimization problems, Bull. Am. math. Soc., 1, 443-474 (1979) · Zbl 0441.49011
[7] Guedda, M.; Veron, L., Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Analysis, 13, 879-902 (1989) · Zbl 0714.35032
[8] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001
[9] Lions, P.-L., The concentration-compactness principle in the calculus of variations: the limit case, Rev. Mat. Ibero., 2, 45-121 (1985) · Zbl 0704.49006
[10] Mancini, G.; Musina, R., A free boundary problem involving limiting Sobolev exponents, Manuscripta math., 58, 77-93 (1987) · Zbl 0601.49004
[11] Mancini, G.; Musina, R., Holes and obstacles, Ann. Inst. H. Poincare Analyse non Lineaire, 5, 323-345 (1988) · Zbl 0666.35039
[12] Noussair, E. S.; Swanson, C. A.; Jianfu, Y., Quasilinear elliptic problems with critical exponents, Nonlinear Analysis, 20, 285-301 (1993) · Zbl 0785.35042
[13] Pohozaev, S., Eigenfunctions of the equation Δ \(u\) + λ \(f(u) = 0\), Dokl. Akad. Nauk. SSSR, 165, 33-36 (1965)
[14] Rodrigues, J. F., Obstacle Problems in Mathematical Physics, Mathematics Studies, 134 (1987), Elsevier: Elsevier the Netherlands · Zbl 0606.73017
[15] Szukin, A., Minimax principle for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincare Analysis non Lineaire, 3, 77-109 (1986)
[16] Jianfu Y., Positive solutions of an obstacle problem, Ann. Fac. Sci. Toulouse (to appear).; Jianfu Y., Positive solutions of an obstacle problem, Ann. Fac. Sci. Toulouse (to appear). · Zbl 0866.49017
[17] Jianfu Y., Regularity of weak solutions to quasilinear elliptic obstacle problems Ann. Fac. Sci. Toulouse (to appear).; Jianfu Y., Regularity of weak solutions to quasilinear elliptic obstacle problems Ann. Fac. Sci. Toulouse (to appear). · Zbl 0877.35023
[18] Xiping, Z., Nontrivial solution of quasilinear elliptic equations involving critical Sobolev exponent, Scientia Sin., 31, 1166-1181 (1988) · Zbl 0677.35039
[19] Xiping, Z.; Jianfu, Y., Quasilinear elliptic equations involving critical Sobolev exponent on unbounded domains, J. Partial diff. Eqns, 2, 53-64 (1989) · Zbl 0694.35062
[20] Jianfu, Y.; Xiping, Z., On the existence of nontrivial solutions of a quasilinear elliptic boundary value problem for unbounded domains (I) and (II), Acta Math. Sci., 7, 447-459 (1987) · Zbl 0697.35051
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