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A priori estimates and existence for quasi-linear elliptic equations. (English) Zbl 1169.35336
Summary: We study the boundary value problem for quasi-linear elliptic equation
\begin{aligned}{\text{div}}(|\nabla u|^{m-2} \nabla u) + B(z,u,\nabla u) &= 0 \quad \text{in }\Omega,\\ u &= 0 \quad \text{on } \partial\Omega, \end{aligned} where $$\Omega\subset\mathbb R^n$$ $$(n\geq 2)$$ is a connected smooth domain, and the exponent $${m\in(1,n)}$$ is a positive number. Under appropriate conditions on the function $$B$$, a variety of results on a priori estimates, existence and non-existence of positive solutions are established. The results are generically optimum for the canonical prototype $$B = |u|^{p-1} u$$, $$p>m-1$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35B45 A priori estimates in context of PDEs 35J20 Variational methods for second-order elliptic equations
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##### References:
 [1] Acerbi E. and Fusco N. (1989). Regularity for minimizers of non-quadratic functionals: the case 1 < p < 2. J. Math. Anal. Appl. 140: 115–135 · Zbl 0686.49004 · doi:10.1016/0022-247X(89)90098-X [2] Dancer, N., Wei, J., Weth, T.: A priori bounds verses multiple existence of positive solutions for a non-linear Schrodinger system. Preprint · Zbl 1191.35121 [3] de Figueiredo D., Lions P. and Naussbaum R. (1982). A priori estimates and existence of positive solutions of emilinear elliptic equations. J. Math. Pures Appl. 61: 41–63 · Zbl 0452.35030 [4] DiBenedetto E. (1983). C 1+$$\alpha$$ Local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7: 827–850 · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5 [5] Gidas B. and Spruck J. (1981). A priori bounds for positive solutions of nonlinear elliptic equations. Commun. PDE 6: 883–901 · Zbl 0462.35041 · doi:10.1080/03605308108820196 [6] Lieberman G.M. (1988). Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12: 1202–1219 · Zbl 0675.35042 · doi:10.1016/0362-546X(88)90053-3 [7] Ruiz D. (2004). A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differ. Equ. 199: 96–114 · Zbl 1081.35037 · doi:10.1016/j.jde.2003.10.021 [8] Serrin J. and Zou H. (2002). Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189: 79–142 · Zbl 1059.35040 · doi:10.1007/BF02392645 [9] Tolksdorf P. (1984). Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51: 126–150 · Zbl 0522.35018 · doi:10.1016/0022-0396(84)90105-0 [10] Vazquez J.-L. (1984). A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12: 191–202 · Zbl 0561.35003 · doi:10.1007/BF01449041 [11] Zou H. (2006). A priori estimates and existence for strongly coupled semilinear cooperative elliptic systems. Commun. PDE 31: 735–773 · Zbl 1137.35027 · doi:10.1080/03605300500394470 [12] Zou H. (2007). Existence and non-existence for strongly coupled quasi-linear cooperative elliptic systems. J. Math. Soc. Jpn. 59(2): 393–421 · Zbl 1197.35099 · doi:10.2969/jmsj/05920393
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