Deng, Yinbin; Jin, Lingyu Multiple positive solutions for a quasilinear nonhomogeneous Neumann problems with critical Hardy exponents. (English) Zbl 1131.35030 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 12, 3261-3275 (2007). 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. Cited in 6 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35B50 Maximum principles in context of PDEs Keywords:Neumann boundary value problems; positive solutions; \(p\)-Laplacian; Hardy equality; quasilinear elliptic problems PDF BibTeX XML Cite \textit{Y. Deng} and \textit{L. Jin}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 12, 3261--3275 (2007; Zbl 1131.35030) Full Text: DOI OpenURL References: [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. 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 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.