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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\(<q<s\).

35J62 Quasilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
Full Text: DOI
[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, (in Chinese)
[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
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