## 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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### References:

 [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 New York · Zbl 0457.35001 [9] Lions, P.-L.; Lions, P.-L., The concentration-compactness principle in the calculus of variations: the limit case, Rev. mat. ibero., 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 δu + λf(u) = 0, Dokl. akad. nauk. SSSR, 165, 33-36, (1965) [14] Rodrigues, J.F., Obstacle problems in mathematical physics, mathematics studies, 134, (1987), Elsevier the Netherlands [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). · Zbl 0866.49017 [17] 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.; 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., Acta math. sci., 7, 447-459, (1987) · Zbl 0697.35051
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