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

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