Weak linking theorems and Schrödinger equations with critical Sobolev exponent

Schechter, Martin; Zou, Wenming

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 601-619.

In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais-Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V\left(x\right)u={K\left(x\right)\left|u\right|}^{2^{*}-2}u+g(x,u),u\in W^{1,2}\left({𝐑}^N\right)$, where $N\ge 4;V,K,g$ are periodic in $x_j$ for $1\le j\le N$ and 0 is in a gap of the spectrum of $-\Delta +V$; $K>0$. If $0<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant $c$, we show that this equation has a nontrivial solution.

Classification : 35B33, 35J65, 35Q55
Mots clés : linking, Schrödinger equations, critical Sobolev exponent

[1] S. Alama and Y.Y. Li, Existence of solutions for semilinear elliptic equations with indefinite linear part. J. Differential Equations 96 (1992) 89-115. | MR | Zbl

[2] S. Alama and Y.Y. Li, On “multibump” bound states for certain semilinear elliptic equations. Indiana J. Math. 41 (1992) 983-1026. | Zbl

[3] T. Bartsch and Y. Ding, On a nonlinear Schrödinger equation with periodic potential. Math. Ann. 313 (1999) 15-37. | MR | Zbl

[4] V. Benci and G. Cerami, Existence of positive solutions of the equation $-\Delta u+a\left(x\right)u=u^{(N+2)/(N-2)}$ in ${𝐑}^N$. J. Funct. Anal. 88 (1990) 90-117. | MR | Zbl

[5] B. Buffoni, L. Jeanjean and C.A. Stuart, Existence of nontrivial solutions to a strongly indefinite semilinear equation. Proc. Amer. Math. Soc. 119 (1993) 179-186. | MR | Zbl

[6] J. Chabrowski and A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent. Preprint of Stockholm University. | Zbl

[7] J. Chabrowski and J. Yang, On Schrödinger equation with periodic potential and critical Sobolev exponent. Topol. Meth. Nonl. Anal. 12 (1998) 245-261. | MR | Zbl

[8] V. Coti-Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${𝐑}^N$. Comm. Pure Appl. Math. 45 (1992) 1217-1269. | Zbl

[9] N. Dunford and J.T. Schwartz, Linear Operators. Part I. Interscience (1967). | Zbl

[10] L. Jeanjean, Solutions in spectral gaps for a nonlinear equation of Schrödinger type. J. Differential Equations 112 (1994) 53-80. | MR | Zbl

[11] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on ${𝐑}^N$. Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787-809. | MR | Zbl

[12] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv. Differential Equations 3 (1998) 441-472. | MR | Zbl

[13] P. Kuchment, Floquet Theory for Partial Differential Equations. Birkhäuser, Basel (1993). | MR | Zbl

[14] Y.Y. Li, On $-\Delta u=K\left(x\right)u^5$ in ${𝐑}^3$. Comm. Pure Appl. Math. 46 (1993) 303-340. | MR

[15] Y.Y. Li, Prescribing scalar curvature on $S^n$ and related problems. Part I. J. Differential Equations 120 (1995) 319-410. | MR | Zbl

[16] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 223-283. | EuDML | Numdam | MR | Zbl

[17] M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV. Academic Press (1978). | MR | Zbl

[18] M. Schechter, Critical point theory with weak-to-weak linking. Comm. Pure Appl. Math. 51 (1998) 1247-1254. | MR | Zbl

[19] M. Schechter, Ratationally invariant periodic solutions of semilinear wave equations. Preprint of the Department of Mathematics, University of California (1998). | MR | Zbl

[20] M. Schechter, Linking Methods in Critical Point Theory. Birkhäuser, Boston (1999). | MR | Zbl

[21] M. Struwe, The existence of surfaces of constant mean curvature with free boundaries. Acta Math. 160 (1988) 19-64. | MR | Zbl

[22] C.A. Stuart, Bifurcation into Spectral Gaps. Bull. Belg. Math. Soc. Suppl. (1995). | MR | Zbl

[23] A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187 (2001) 25-41. | MR | Zbl

[24] C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation. Comm. Partial Differential Equations 21 (1996) 1431-1449. | MR | Zbl

[25] M. Willem and W. Zou, On a semilinear Dirichlet problem and a nonlinear Schrödinger equation with periodic potential. Indiana Univ. Math. J. 52 (2003) 109-132. | MR | Zbl

[26] M. Willem, Minimax Theorems. Birkhäuser, Boston (1996). | MR | Zbl

[27] W. Zou, Solitary Waves of the Generalized Kadomtsev-Petviashvili Equations. Appl. Math. Lett. 15 (2002) 35-39. | MR | Zbl

[28] W. Zou, Variant Fountain Theorems and their Applications. Manuscripta Math. 104 (2001) 343-358. | MR | Zbl

[29] M. Schechter, Some recent results in critical point theory. Pan Amer. Math. J. 12 (2002) 1-19. | MR | Zbl

[30] M. Schechter and W. Zou, Homoclinic Orbits for Schrödinger Systems. Michigan Math. J. 51 (2003) 59-71. | MR | Zbl

[31] M. Schechter and W. Zou, Superlinear Problem. Pacific J. Math. (accepted). | MR | Zbl

[32] W. Zou and S. Li, New Linking Theorem and Elliptic Systems with Nonlinear Boundary Condition. Nonl. Anal. TMA 52 (2003) 1797-1820. | MR | Zbl