Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting. (English) Zbl 1176.35071
Summary: We study the boundary value problem
\[
-\text{div}(\log(1+|\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)\;\text{in}\;\Omega,\quad u=0\;\text{on}\;\partial\Omega,
\]
where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary. We distinguish the cases where either \(f(u)=-\lambda |u|^{p-2}u+|u|^{r-2}u\) or \(f(u)=\lambda |u|^p-2u-|u|^{r-2}u\), with \(p, q>1\), \(p+q<\min\{N,r\}\), and \(r<(Np-N+p)/(N-p)\). In the first case we show the existence of infinitely many weak solutions for any \(\lambda>0\). In the second case we prove the existence of a nontrivial weak solution if \(\lambda\) is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
References:
| [1] | Acerbi, E.; Mingione, G., Gradient estimates for the \(p(x)\)-Laplacean system, J. Reine Angew. Math., 584, 117-148 (2005) · Zbl 1093.76003 |
| [2] | Adams, R., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030 |
| [3] | Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 |
| [4] | Benci, V.; Fortunato, D., Discreteness conditions of the spectrum of Schrödinger operators, J. Math. Anal. Appl., 64, 695-700 (1978) · Zbl 0389.35016 |
| [5] | Chabrowski, J.; Fu, Y., Existence of solutions for \(p(x)\)-Laplacian problems on a bounded domain, J. Math. Anal. Appl., 306, 604-618 (2005) · Zbl 1160.35399 |
| [6] | Chen, Y.; Levine, S.; Rao, R., Functionals with \(p(x)\)-growth in image processing, Duquesne University, Department of Mathematics and Computer Science Technical Report 2004-01, available at |
| [7] | Clément, Ph.; García-Huidobro, M.; Manásevich, R.; Schmitt, K., Mountain pass type solutions for quasilinear elliptic equations, Calc. Var., 11, 33-62 (2000) · Zbl 0959.35057 |
| [8] | Clément, Ph.; de Pagter, B.; Sweers, G.; de Thélin, F., Existence of solutions to a semilinear elliptic system through Orlicz-Sobolev spaces, Mediterr. J. Math., 1, 241-267 (2004) · Zbl 1167.35352 |
| [9] | G. Dankert, Sobolev embedding theorems in Orlicz spaces, PhD thesis, University of Köln, 1966; G. Dankert, Sobolev embedding theorems in Orlicz spaces, PhD thesis, University of Köln, 1966 |
| [10] | Donaldson, T. K., Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces, J. Differential Equations, 10, 507-528 (1971) · Zbl 0207.41501 |
| [11] | Donaldson, T. K.; Trudinger, N. S., Orlicz-Sobolev spaces and imbedding theorems, J. Funct. Anal., 8, 52-75 (1971) · Zbl 0216.15702 |
| [12] | Fan, X., Solutions for \(p(x)\)-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl., 312, 464-477 (2005) · Zbl 1154.35336 |
| [13] | Fan, X.; Zhang, Q.; Zhao, D., Eigenvalues of \(p(x)\)-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302, 306-317 (2005) · Zbl 1072.35138 |
| [14] | García-Huidobro, M.; Le, V. K.; Manásevich, R.; Schmitt, K., On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting, Nonlinear Differential Equations Appl., 6, 207-225 (1999) · Zbl 0936.35067 |
| [15] | Gossez, J. P., Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 190, 163-205 (1974) · Zbl 0239.35045 |
| [16] | Gossez, J. P., A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, (Proc. Sympos. Pure Math., vol. 45 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 455-462 · Zbl 0602.35037 |
| [17] | Le, V. K.; Schmitt, K., Quasilinear elliptic equations and inequalities with rapidly growing coefficients, J. London Math. Soc., 62, 852-872 (2000) · Zbl 1013.35032 |
| [18] | W. Luxemburg, Banach function spaces, PhD thesis, Technische Hogeschool te Delft, The Netherlands, 1955; W. Luxemburg, Banach function spaces, PhD thesis, Technische Hogeschool te Delft, The Netherlands, 1955 · Zbl 0068.09204 |
| [19] | Krasnosel’skii, M. A.; Rutickii, Ya. B., Convex Functions and Orlicz Spaces (1961), Noordhoff: Noordhoff Gröningen · Zbl 0095.09103 |
| [20] | Kufner, A.; John, O.; Fučik, S., Function Spaces (1997), Noordhoff: Noordhoff Leyden |
| [21] | Mihăilescu, M.; Rădulescu, V., A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. Roy. Soc. London Ser. A, 462, 2625-2641 (2006) · Zbl 1149.76692 |
| [22] | O’Neill, R., Fractional integration in Orlicz spaces, Trans. Amer. Math. Soc., 115, 300-328 (1965) · Zbl 0132.09201 |
| [23] | Rabinowitz, P., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. (1984), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI |
| [24] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics, vol. IV, Analysis of Operators (1978), Academic Press: Academic Press New York · Zbl 0401.47001 |
| [25] | Ružička, M., Electrorheological Fluids: Modeling and Mathematical Theory (2000), Springer: Springer Berlin · Zbl 0968.76531 |
| [26] | Struwe, M., Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (1996), Springer: Springer Heidelberg · Zbl 0864.49001 |
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