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Existence and multiplicity of positive solutions for singular quasilinear problems. (English) Zbl 1168.35358
Summary: We combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular $$p$$-Laplacian problems $\begin{cases} -\Delta_ pu=a(x)u^ {-\gamma}+\lambda f(x,u)\quad &\text{ in }\Omega,\\ \quad u\in W_ {0}^ {1,p}(\Omega),\;u>0\quad &\text{ in }\Omega, \end{cases}\tag{1}$ where $$\Omega \subset \mathbb{R}^ {N}$$, $$N\geq 1$$, is a $$C^ {2}$$ bounded domain, $$1<p<\infty$$, $$\gamma >0$$ is a constant, $$\lambda >0$$ is a parameter, $$f$$ is a Carathéodory function and $$a\geq 0$$ is measurable and satisfies $$\varphi^ {-\gamma}a\in L^ q(\Omega)$$ for some $$q>N$$ and $$\varphi\in C_ 0^ 1(\overline{\Omega})$$, $$\varphi \geq 0$$.
In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.

##### MSC:
 35J62 Quasilinear elliptic equations 35B32 Bifurcations in context of PDEs 35B09 Positive solutions to PDEs 35D30 Weak solutions to PDEs 35J20 Variational methods for second-order elliptic equations 47J30 Variational methods involving nonlinear operators
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##### References:
  Agarwal, R.P.; Lü, H.; O’Regan, D., Existence theorems for the one-dimensional singular p-Laplacian equation with sign changing nonlinearities, Appl. math. comput., 143, 1, 15-38, (2003) · Zbl 1031.34023  R.P. Agarwal, K. Perera, D. O’Regan, A variational approach to singular quasilinear elliptic problems with sign changing nonlinearities, Appl. Anal., in press  Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063  Brezis, H.; Nirenberg, L., $$H^1$$ versus $$C^1$$ local minimizers, C. R. acad. sci. Paris Sér. I math., 317, 5, 465-472, (1993) · Zbl 0803.35029  Coclite, M.M.; Palmieri, G., On a singular nonlinear Dirichlet problem, Comm. partial differential equations, 14, 10, 1315-1327, (1989) · Zbl 0692.35047  Crandall, M.G.; Rabinowitz, P.H.; Tartar, L., On a Dirichlet problem with a singular nonlinearity, Comm. partial differential equations, 2, 2, 193-222, (1977) · Zbl 0362.35031  del Pino, M.A., A global estimate for the gradient in a singular elliptic boundary value problem, Proc. roy. soc. Edinburgh sect. A, 122, 3-4, 341-352, (1992) · Zbl 0791.35046  Damascelli, L., Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. inst. H. Poincaré anal. non linéaire, 15, 4, 493-516, (1988) · Zbl 0911.35009  Diaz, J.I.; Morel, J.-M.; Oswald, L., An elliptic equation with singular nonlinearity, Comm. partial differential equations, 12, 12, 1333-1344, (1987) · Zbl 0634.35031  Edelson, A.L., Entire solutions of singular elliptic equations, J. math. anal. appl., 139, 2, 523-532, (1989) · Zbl 0679.35003  Guo, Z.; Zhang, Z., $$W^{1, p}$$ versus $$C^1$$ local minimizers and multiplicity results for quasilinear elliptic equations, J. math. anal. appl., 286, 1, 32-50, (2003) · Zbl 1160.35382  Kusano, T.; Swanson, C.A., Entire positive solutions of singular semilinear elliptic equations, Japan. J. math. (N.S.), 11, 1, 145-155, (1985) · Zbl 0585.35034  Lair, A.V.; Shaker, A.W., Classical and weak solutions of a singular semilinear elliptic problem, J. math. anal. appl., 211, 2, 371-385, (1997) · Zbl 0880.35043  Lazer, A.C.; McKenna, P.J., On a singular nonlinear elliptic boundary-value problem, Proc. amer. math. soc., 111, 3, 721-730, (1991) · Zbl 0727.35057  Perera, K.; Zhang, Z., Multiple positive solutions of singular p-Laplacian problems by variational methods, Boundary value problems, 3, 377-382, (2005) · Zbl 1220.35082  Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, CBMS reg. conf. ser. math., vol. 65, (1986), published for the Conference Board of the Mathematical Sciences Washington, DC · Zbl 0609.58002  Shaker, A.W., On singular semilinear elliptic equations, J. math. anal. appl., 173, 1, 222-228, (1993) · Zbl 0785.35032  Shi, J.; Yao, M., On a singular nonlinear semilinear elliptic problem, Proc. roy. soc. Edinburgh sect. A, 128, 6, 1389-1401, (1998) · Zbl 0919.35044  Sun, Y.; Wu, S.; Long, Y., Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. differential equations, 176, 2, 511-531, (2001) · Zbl 1109.35344  Vázquez, J.L., A strong maximum principle for some quasilinear elliptic equations, Appl. math. optim., 12, 3, 191-202, (1984) · Zbl 0561.35003  Wiegner, M., A degenerate diffusion equation with a nonlinear source term, Nonlinear anal., 28, 12, 1977-1995, (1997) · Zbl 0874.35061  Zhang, Z., Critical points and positive solutions of singular elliptic boundary value problems, J. math. anal. appl., 302, 2, 476-483, (2005) · Zbl 1161.35403
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