EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

Title & Authors
EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS
Zou, Henghui;

Abstract
We study existence of positive solutions of the classical nonlinear Schr$\small{\ddot{o}}$dinger equation $\small{-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0}$, u > 0 in $\small{\mathbb{R}^n}$ $\small{u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}}$. In fact, we consider the following more general quasi-linear Schr$\small{\ddot{o}}$odinger equation $\small{-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}}$ $\small{-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0}$, u > 0 in $\small{\mathbb{R}^n}$ $\small{u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}}$, where m $\small{\in}$ (1, n) is a positive number and $\small{m^*\;:=\;\frac{mn}{n-m}\;}$>$\small{\;0}$, is the corresponding critical Sobolev embedding number in $\small{\mathbb{R}^n}$. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.
Keywords
concentration-compactness;critical Sobolev exponent;existence;m-Laplacian;minimax methods;mountain-pass lemma;Schr$\small{\ddot{o}}$dinger equations;
Language
English
Cited by
1.
Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in ℝN, Journal of Mathematical Physics, 2016, 57, 11, 111505
2.
Quasilinear scalar field equations with competing potentials, Journal of Differential Equations, 2011, 251, 12, 3625
3.
Existence of solutions for Kirchhoff type problems with critical nonlinearity in $${\mathbb{R}^N}$$ R N, Zeitschrift für angewandte Mathematik und Physik, 2015, 66, 3, 547
4.
Existence of solutions for Kirchhoff type problems with critical nonlinearity in R3, Nonlinear Analysis: Real World Applications, 2014, 17, 126
5.
Existence of solutions for a class of biharmonic equations with critical nonlinearity in $$\mathbb {R}^N$$ R N, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2016, 110, 2, 681
References
1.
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381.

2.
T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag, New York, 1982.

3.
M.-F. Bidaut-Veron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math. 84 (2001), 1-49.

4.
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437-477.

5.
E. DiBenedetto, $C^{1+{\alpha}}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), no. 8, 827-850.

6.
Y. Ding and F. Lin, Solutions of perturbed Schrodinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations 30 (2007), no. 2, 231-249.

7.
P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145-201.

8.
P. L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45-121.

9.
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J. 35 (1986), no. 3, 681-703.

10.
P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.

11.
J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302.

12.
J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), no. 1, 79-142.

13.
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372.

14.
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126-150.

15.
N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747.

16.
J.-L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191-202.

17.
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 24, Birkhauser Boston, Inc., Boston, MA, 1996.