A variational inequality theory for demicontinuous \(S\)-contractive maps with applications to semilinear elliptic inequalities. (English) Zbl 1160.49009
Summary: A variational inequality theory for demicontinuous \(S\)-contractive maps in Hilbert spaces is established by employing the ideas of Granas’ topological transversality. Such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward \(S\)-contractive maps and to those of fixed point index for condensing maps. The variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear second-order elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical Sobolev exponent. There has been little discussion for such elliptic inequalities involving the critical Sobolev exponent in the literature.
MSC:
| 49J40 | Variational inequalities |
| 35R45 | Partial differential inequalities and systems of partial differential inequalities |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 47H05 | Monotone operators and generalizations |
| 47H06 | Nonlinear accretive operators, dissipative operators, etc. |
Keywords:
variational inequality; demicontinuous \(S\)-contractive map; essential map; positive solution; semilinear elliptic inequalitiesReferences:
| [1] | Adams, R. A.; Fournier, J. F., Sobolev Spaces (2003), Academic Press: Academic Press New York · Zbl 1098.46001 |
| [2] | Adhikari, D. R.; Kartsatos, A. G., Topological degree theories and nonlinear operator equations in Banach spaces, Nonlinear Anal., 69, 1235-1255 (2008) · Zbl 1142.47346 |
| [3] | Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044 |
| [4] | Benci, V., Positive solutions of some eigenvalue problems in the theory of variational inequalities, J. Math. Anal. Appl., 61, 165-187 (1977) · Zbl 0432.35059 |
| [5] | Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36, 437-477 (1983) · Zbl 0541.35029 |
| [6] | Browder, F. E., Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc., 9, 1-39 (1983) · Zbl 0533.47053 |
| [7] | Deimling, K., Nonlinear Functional Analysis (1985), Springer: Springer Berlin · Zbl 0559.47040 |
| [8] | Dugundji, J.; Granas, A., Fixed Point Theory, vol. 1 (1982), Polish Scientific Publishers: Polish Scientific Publishers Warsaw · Zbl 0483.47038 |
| [9] | Fang, M., Degenerate elliptic inequalities with critical growth, J. Differential Equations, 232, 441-467 (2007) · Zbl 1166.35334 |
| [10] | Fitzpatrick, P. M.; Petryshyn, W. V., On the nonlinear eigenvalue problem \(T(u) = \lambda C(u)\), involving noncompact abstract and differential operators, Boll. Unione Mat. Ital. Sez. B, 15, 80-107 (1978) · Zbl 0386.47033 |
| [11] | Friedman, A., Variational Principles and Free Boundary Value Problems (1983), Wiley-Interscience: Wiley-Interscience New York |
| [12] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer Berlin · Zbl 0691.35001 |
| [13] | A. Granas, R.B. Guenther, J.W. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertations Mathematicae, CCXLIV, Polska Akademia Nauk., 1985; A. Granas, R.B. Guenther, J.W. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertations Mathematicae, CCXLIV, Polska Akademia Nauk., 1985 · Zbl 0615.34010 |
| [14] | Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045 |
| [15] | Heinonen, J.; Kilpeläinen, T.; Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations (1993), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 0780.31001 |
| [16] | Istratescu, V. I., Fixed Point Theory (1981), Reidel Publ. Comp.: Reidel Publ. Comp. Dordrecht, Holland · Zbl 0465.47035 |
| [17] | Karamardian, S., Generalized complementarity problem, J. Optim. Theory Appl., 8, 161-168 (1971) · Zbl 0218.90052 |
| [18] | Karamardian, S., The complementarity problem, Math. Program., 2, 107-109 (1972) · Zbl 0247.90058 |
| [19] | Kinderlehrer, D.; Stampacchia, G., Introduction to Variational Inequalities and Their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001 |
| [20] | Kirk, W. A., On nonlinear mappings of strongly semicontractive type, J. Math. Anal. Appl., 27, 409-412 (1969) · Zbl 0183.15103 |
| [21] | Krasnosels’kii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations (1963), Pergamon Press: Pergamon Press Oxford |
| [22] | Lan, K. Q., A fixed point theory for weakly inward \(S\)-contractive maps, Nonlinear Anal., 45, 189-201 (2001) · Zbl 0984.47039 |
| [23] | Lan, K. Q., Positive solutions of semi-positone Hammerstein integral equations and applications, Commun. Pure Appl. Anal., 6, 2, 441-451 (2007) · Zbl 1134.45005 |
| [24] | Lan, K. Q., Multiple positive solutions of semi-positone Sturm-Liouville boundary value problems, Bull. London Math. Soc., 38, 283-293 (2006) · Zbl 1094.34014 |
| [25] | Lan, K. Q., Multiple positive solutions of semilinear differential equations with singularities, J. London Math. Soc. (2), 63, 690-704 (2001) · Zbl 1032.34019 |
| [26] | Lan, K. Q.; Webb, J. R.L., Variational inequalities and fixed point theorems for PM-maps, J. Math. Anal. Appl., 224, 102-116 (1998) · Zbl 0942.49011 |
| [27] | Lan, K. Q.; Webb, J. R.L., A fixed point index for generalized inward mappings of condensing type, Trans. Amer. Math. Soc., 349, 2175-2186 (1997) · Zbl 0878.47045 |
| [28] | Lan, K. Q.; Webb, J. R.L., Positive solutions of semilinear differential equations with singularities, J. Differential Equations, 148, 407-421 (1998) · Zbl 0909.34013 |
| [29] | Le, V. K.; Schmitt, K., On boundary value problems for degenerate quasilinear elliptic equations and inequalities, J. Differential Equations, 144, 170-208 (1998) · Zbl 0912.35069 |
| [30] | Lions, J., Optimal Control of Systems Governed by Partial Differential Equations (1971), Springer: Springer New York · Zbl 0203.09001 |
| [31] | Mancino, O.; Stampacchia, G., Convex programming and variational inequalities, J. Optim. Theory Appl., 9, 3-23 (1972) · Zbl 0223.90031 |
| [32] | Nussbaum, R. D., The fixed point index for local condensing maps, Ann. Mat. Pura Appl., 84, 217-258 (1971) · Zbl 0226.47031 |
| [33] | Padilla, P., The effect of the shape of the domain on the existence of solutions of an equation involving the critical Sobolev exponent, J. Differential Equations, 124, 446-471 (1996) · Zbl 0838.35042 |
| [34] | Pohozaev, I., Eigenfunctions of the equation \(\Delta u + f(u) = 0\), Dokl. Akad. Nauk SSSR, 165, 33-36 (1965) |
| [35] | Petryshyn, W. V., Nonlinear eigenvalue problems and the existence of nonzero fixed points for \(A\)-proper mappings, (Kluge, R., Theory of Nonlinear Operators (1978), Akademie-Verlag: Akademie-Verlag Berlin), 215-227 · Zbl 0443.47060 |
| [36] | Petryshyn, W. V., On the solvability of \(x \in T x + \lambda F x\) in quasinormal cones with \(T\) and F k-set-contractive, Nonlinear Anal., 5, 585-591 (1981) · Zbl 0474.47028 |
| [37] | Precup, R., On the topological transversality principle, Nonlinear Anal., 20, 1-9 (1993) · Zbl 0780.47048 |
| [38] | Precup, R., On some fixed point theorems of Deimling, Nonlinear Anal., 23, 1315-1320 (1994) · Zbl 0856.47034 |
| [39] | Skrypnik, I. V., Nonlinear Higher Order Elliptic Equations (1973), Naukova Dumka: Naukova Dumka Kiev, (in Russian) · Zbl 0296.35032 |
| [40] | Skrypnik, I. V., Methods for Analysis of Nonlinear Elliptic Boundary Value Problems, Amer. Math. Soc. Transl. Ser. 2, vol. 139 (1994), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0822.35001 |
| [41] | Szulkin, A., Positive solutions of variational inequalities: A degree-theoretic approach, J. Differential Equations, 57, 90-111 (1985) · Zbl 0535.35029 |
| [42] | Webb, J. R.L.; Lan, K. Q., Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type, Topol. Methods Nonlinear Anal., 27, 1, 91-116 (2006) · Zbl 1146.34020 |
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