Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions. (English) Zbl 1222.35090
Summary: We consider the following quasilinear elliptic exterior problem
\[ \begin{cases} -\text{div}\big(a(x)|\nabla u|^{p-2}\nabla u\big)+ g(x)|u|^{q-2}u= h(x)|u|^{s-2}u+ \lambda H(x)|u|^{r-2}u, &x\in\Omega,\\ a(x)|\nabla u|^{p-2} \frac{\partial u}{\partial\nu}+ b(x)|u|^{p-2}u=0, &x\in\Gamma=\partial\Omega, \end{cases} \]
where \(\Omega\) is a smooth exterior domain in \(\mathbb R^N\), and \(\nu\) is the unit vector of the outward normal on the boundary \(\Gamma\), \(1<p<N\), \(1<s<p<r<p^*= Np/(N-p)\). By the variational principle and the mountain pass theorem, we establish the existence of infinitely many solutions if \(q>r\) and at least one solution if 1\(<q<s\).
\[ \begin{cases} -\text{div}\big(a(x)|\nabla u|^{p-2}\nabla u\big)+ g(x)|u|^{q-2}u= h(x)|u|^{s-2}u+ \lambda H(x)|u|^{r-2}u, &x\in\Omega,\\ a(x)|\nabla u|^{p-2} \frac{\partial u}{\partial\nu}+ b(x)|u|^{p-2}u=0, &x\in\Gamma=\partial\Omega, \end{cases} \]
where \(\Omega\) is a smooth exterior domain in \(\mathbb R^N\), and \(\nu\) is the unit vector of the outward normal on the boundary \(\Gamma\), \(1<p<N\), \(1<s<p<r<p^*= Np/(N-p)\). By the variational principle and the mountain pass theorem, we establish the existence of infinitely many solutions if \(q>r\) and at least one solution if 1\(<q<s\).
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
existence; quasilinear elliptic equation; exterior domain; nonlinear boundary condition; concave and convex nonlinearitiesReferences:
| [1] | Afrouzi, G. A.; Rasouli, S. H., A variational approach to a quasilinear elliptic problem involving the \(p\)-Laplacian and nonlinear boundary conditions, Nonlinear Anal., 71, 2447-2455 (2009) · Zbl 1173.35487 |
| [2] | Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 |
| [3] | Atkinson, C.; Kalli, K. E., Some boundary value problems for the Bingham model, J. Non-Newtonian Fluid Mech., 41, 339-363 (1992) · Zbl 0747.76012 |
| [4] | Brown, K. J.; Wu, T. F., A semilinear elliptic system involving nonlinear boundary condition and sign-changing weighted function, J. Math. Anal. Appl., 337, 1326-1336 (2008) · Zbl 1132.35361 |
| [5] | Cîrstea, F.; Motreanu, D.; Rădulescu, V., Weak solutions of quasilinear problems with nonlinear boundary condition, Nonlinear Anal., 43, 623-636 (2001) · Zbl 0972.35038 |
| [6] | Diaz, J. I., Nonlinear Partial Differential Equations and Free Boundaries (1985), Pitman Publ. Program · Zbl 0595.35100 |
| [7] | Escobar, J. F., Uniqueness theorems on conformal deformations of metrics, Sobolev inequalities, and an eigenvalue estimate, Comm. Pure Appl. Math., 43, 857-883 (1990) · Zbl 0713.53024 |
| [8] | Filippucci, R.; Pucci, P.; Rădulescu, V., Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations, 33, 706-717 (2008) · Zbl 1147.35038 |
| [9] | Ilʼyasov, Y.; Runst, T., On nonlocal calculation for inhomogeneous indefinite Neumann boundary value problems, Calc. Var., 22, 101-127 (2005) · Zbl 1161.35392 |
| [10] | Kang, K. C., Critical Point Theory and Its Applications (1986), Shanghai Science Tech. Press: Shanghai Science Tech. Press Shanghai, (in Chinese) · Zbl 0698.58002 |
| [11] | Keller, E. F.; Segel, L. A., Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26, 399-415 (1970) · Zbl 1170.92306 |
| [12] | Liu, S. B.; Li, S. J., An elliptic equation with concave and convex nonlinearities, Nonlinear Anal., 53, 723-731 (2003) · Zbl 1217.35067 |
| [13] | Santos, C. A., Nonexistence and existence of entire solutions for a quasilinear problem with singular and super-linear terms, Nonlinear Anal., 72, 3813-3819 (2010) · Zbl 1189.35104 |
| [14] | Struwe, M., Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (2000), Springer · Zbl 0939.49001 |
| [15] | Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51, 126-150 (1984) · Zbl 0488.35017 |
| [16] | Tonkes, E., A semilinear elliptic equation with convex and concave nonlinearities, Topol. Methods Nonlinear Anal., 13, 251-271 (1999) · Zbl 0991.35022 |
| [17] | Tshinanga, S. B., On multiple solutions of semilinear elliptic equation on unbounded domains with concave and convex nonlinearities, Nonlinear Anal., 28, 809-814 (1997) · Zbl 0865.35050 |
| [18] | Yu, L. S., Nonlinear \(p\)-Laplacian problem on unbounded domains, Proc. Amer. Math. Soc., 115, 1037-1045 (1992) · Zbl 0754.35036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
