# American Institute of Mathematical Sciences

August  2012, 5(4): 819-830. doi: 10.3934/dcdss.2012.5.819

## A variational approach to a class of quasilinear elliptic equations not in divergence form

 1 Dip. di Matematica, Università di Roma "Tor Vergata", Via della Ricerca Scientifica - 00133 - Roma 2 Dipartimento di Matematica, Università della Calabria, Ponte Pietro Bucci 31 B, Arcavacata di Rende (Cosenza), 87036, Italy

Received  February 2011 Revised  May 2011 Published  November 2011

The aim of this paper is to use a variational approach in order to obtain the existence of non-trivial weak solutions of a quasilinear elliptic equation not in divergence form, in dimension $N=3$. Moreover, we prove that our solution is $C^{1, \alpha}(\overline\Omega)$ and also locally $C^{2, \alpha}(\overline\Omega)$ for a suitable $\alpha\in (0,1)$.
Citation: M. Matzeu, Raffaella Servadei. A variational approach to a class of quasilinear elliptic equations not in divergence form. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 819-830. doi: 10.3934/dcdss.2012.5.819
##### References:
 [1] H. Amann and M. G. Crandall, On some existence theorems for semi-linear elliptic equations,, Indiana Univ. Math. J., 27 (1978), 779. doi: 10.1512/iumj.1978.27.27050. Google Scholar [2] A. Ambrosetti and D. Arcoya, On a quasilinear problem at strong resonance,, Topol. Methods Nonlinear Anal., 6 (1995), 255. Google Scholar [3] A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems,", Cambridge Studies in Advanced Mathematics, 104 (2007). Google Scholar [4] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [5] D. Arcoya and P. J. Martinez-Aparicio, Quasilinear equations with natural growth,, Rev. Mat. Iberoam., 24 (2008), 597. Google Scholar [6] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms,, ESAIM Control Optim. Calc. Var., 14 (2008), 411. doi: 10.1051/cocv:2008031. Google Scholar [7] L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data,, Nonlinear Anal., 19 (1992), 573. doi: 10.1016/0362-546X(92)90022-7. Google Scholar [8] H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems,, Comm. Partial Differential Equations, 2 (1977), 601. Google Scholar [9] D. De Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques,, Differential Integral Equations, 17 (2004), 119. Google Scholar [10] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", reprint of the 1998 edition, (1998). Google Scholar [12] M. Girardi and M. Matzeu, Positive and negative solutions of a quasi-linear elliptic equation by a Mountain Pass method and truncature techniques,, Nonlinear Anal., 59 (2004), 199. Google Scholar [13] M. Girardi and M. Matzeu, A compactness result for quasilinear elliptic equations by Mountain Pass techniques,, Rend. Mat. Appl. (7), 29 (2009), 83. Google Scholar [14] S. Pohožaev, Equations of the type $\Delta u=f(x,u,Du)$,, Mat. Sb. (N.S.), 113(155) (1980), 324. Google Scholar [15] R. Servadei, A semilinear elliptic PDE not in divergence form via variational methods,, J. Math. Anal. Appl., (). Google Scholar [16] J. B. M. Xavier, Some existence theorems for equations of the form $-\Delta u=f(x,u,Du)$,, Nonlinear Anal., 15 (1990), 59. Google Scholar [17] Z. Yan, A note on the solvability in $W^{2, p}(\Omega)$ for the equation $-\Delta u=f(x,u,Du)$,, Nonlinear Anal., 24 (1995), 1413. Google Scholar

show all references

##### References:
 [1] H. Amann and M. G. Crandall, On some existence theorems for semi-linear elliptic equations,, Indiana Univ. Math. J., 27 (1978), 779. doi: 10.1512/iumj.1978.27.27050. Google Scholar [2] A. Ambrosetti and D. Arcoya, On a quasilinear problem at strong resonance,, Topol. Methods Nonlinear Anal., 6 (1995), 255. Google Scholar [3] A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems,", Cambridge Studies in Advanced Mathematics, 104 (2007). Google Scholar [4] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [5] D. Arcoya and P. J. Martinez-Aparicio, Quasilinear equations with natural growth,, Rev. Mat. Iberoam., 24 (2008), 597. Google Scholar [6] L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms,, ESAIM Control Optim. Calc. Var., 14 (2008), 411. doi: 10.1051/cocv:2008031. Google Scholar [7] L. Boccardo and T. Gallouët, Strongly nonlinear elliptic equations having natural growth terms and $L^1$ data,, Nonlinear Anal., 19 (1992), 573. doi: 10.1016/0362-546X(92)90022-7. Google Scholar [8] H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems,, Comm. Partial Differential Equations, 2 (1977), 601. Google Scholar [9] D. De Figueiredo, M. Girardi and M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques,, Differential Integral Equations, 17 (2004), 119. Google Scholar [10] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar [11] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", reprint of the 1998 edition, (1998). Google Scholar [12] M. Girardi and M. Matzeu, Positive and negative solutions of a quasi-linear elliptic equation by a Mountain Pass method and truncature techniques,, Nonlinear Anal., 59 (2004), 199. Google Scholar [13] M. Girardi and M. Matzeu, A compactness result for quasilinear elliptic equations by Mountain Pass techniques,, Rend. Mat. Appl. (7), 29 (2009), 83. Google Scholar [14] S. Pohožaev, Equations of the type $\Delta u=f(x,u,Du)$,, Mat. Sb. (N.S.), 113(155) (1980), 324. Google Scholar [15] R. Servadei, A semilinear elliptic PDE not in divergence form via variational methods,, J. Math. Anal. Appl., (). Google Scholar [16] J. B. M. Xavier, Some existence theorems for equations of the form $-\Delta u=f(x,u,Du)$,, Nonlinear Anal., 15 (1990), 59. Google Scholar [17] Z. Yan, A note on the solvability in $W^{2, p}(\Omega)$ for the equation $-\Delta u=f(x,u,Du)$,, Nonlinear Anal., 24 (1995), 1413. Google Scholar
 [1] Ilaria Fragalà, Filippo Gazzola, Gary Lieberman. Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains. Conference Publications, 2005, 2005 (Special) : 280-286. doi: 10.3934/proc.2005.2005.280 [2] Dung Le. Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 877-893. doi: 10.3934/dcdsb.2017044 [3] Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198 [4] Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043 [5] Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043 [6] Enrico Gerlach, Charlampos Skokos. Comparing the efficiency of numerical techniques for the integration of variational equations. Conference Publications, 2011, 2011 (Special) : 475-484. doi: 10.3934/proc.2011.2011.475 [7] Jaime Arango, Adriana Gómez. Critical points of solutions to elliptic problems in planar domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 327-338. doi: 10.3934/cpaa.2011.10.327 [8] Fengping Yao, Shulin Zhou. Interior $C^{1,\alpha}$ regularity of weak solutions for a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1635-1649. doi: 10.3934/dcdsb.2016015 [9] Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016 [10] Brahim Bougherara, Jacques Giacomoni, Jesus Hernández. Some regularity results for a singular elliptic problem. Conference Publications, 2015, 2015 (special) : 142-150. doi: 10.3934/proc.2015.0142 [11] Guji Tian, Xu-Jia Wang. Partial regularity for elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 899-913. doi: 10.3934/dcds.2010.28.899 [12] Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25. [13] Sun-Sig Byun, Hongbin Chen, Mijoung Kim, Lihe Wang. Lp regularity theory for linear elliptic systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 121-134. doi: 10.3934/dcds.2007.18.121 [14] Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477 [15] Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133 [16] Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179 [17] Marcos L. M. Carvalho, José Valdo A. Goncalves, Claudiney Goulart, Olímpio H. Miyagaki. Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 83-106. doi: 10.3934/cpaa.2019006 [18] Filippo Gazzola. Critical exponents which relate embedding inequalities with quasilinear elliptic problems. Conference Publications, 2003, 2003 (Special) : 327-335. doi: 10.3934/proc.2003.2003.327 [19] Claudianor Oliveira Alves, Paulo Cesar Carrião, Olímpio Hiroshi Miyagaki. Signed solution for a class of quasilinear elliptic problem with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (4) : 531-545. doi: 10.3934/cpaa.2002.1.531 [20] Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105

2018 Impact Factor: 0.545