Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents. (English) Zbl 1131.35030

Summary: We study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problems of the type
\[ \begin{aligned} -\Delta_pu+\lambda u^{p-1}= \frac{u^q}{|x|^s}, &\quad x\in\Omega,\\ u>0, &\quad x\in\Omega,\\ |\nabla u|^{p-2}D_\gamma u=\varphi, &\quad x\in\partial\Omega \setminus\{0\}, \end{aligned} \]
where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with smooth boundary, \(0\in\partial\Omega\), \(2\leq p<n\). \(\Delta_pu= \text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(p-1<q\leq p^*(s)-1\), \(0\leq s<p-1\), \(p^*(s)= \frac{(n-s)p}{n-p}\), \(\varphi\in C^\alpha (\overline{\Omega})\), \(0<\alpha<1\), \(\varphi(x)\geq 0\), \(\varphi(x)\not\equiv 0\) and \(\lambda\) is a real constant.


35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
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[1] Garcia Azorero, J. P.; Alonso, I. P., Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144, 441-476 (1998) · Zbl 0918.35052
[2] Abreu, E. A.M.; do Ó, J. M.; Medeiros, E. S., Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems, Nonlinear Anal., 60, 1443-1471 (2005) · Zbl 1151.35366
[3] 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
[4] Deng, Y. B.; Peng, S. J., Exitence of multiple solutions for inhomogeneous Neumann problem, J. Math. Anal. Appl., 271, 155-174 (2002)
[5] Ding, W. Y.; Ni, W. M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91, 283-308 (1986) · Zbl 0616.35029
[6] Ghoussoub, N.; Yuan, C., Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352, 5703-5743 (2000) · Zbl 0956.35056
[7] Tolksdorf, P., On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, 8, 773-817 (1983) · Zbl 0515.35024
[8] Willem, M., Minimax Theorems (1996), Birkhuser Boston, Inc.: Birkhuser Boston, Inc. Boston · Zbl 0856.49001
[9] Wang, X. J., Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93, 283-310 (1991) · Zbl 0766.35017
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