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 $-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\; Keywords concentration-compactness;critical Sobolev exponent;existence;m-Laplacian;minimax methods;mountain-pass lemma;Schr$\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.

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