# 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] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [2] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [3] Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051 [4] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 [5] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469 [6] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [7] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [8] Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033 [9] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [10] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [11] Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115 [12] Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451 [13] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [14] Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170 [15] Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032 [16] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [17] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [18] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453 [19] Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458 [20] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

2019 Impact Factor: 1.233