Generalized linking theorem with an application to a semilinear Schrödinger equation

Consider the semilinear Schrodinger equation (*) $-\Delta u + V(x)u = f(x,u)$, $u\in H^1(\mathbf {R} ^N)$. It is shown that if $f$, $V$ are periodic in the $x$-variables, $f$ is superlinear at $u=0$ and $\pm\infty$ and 0 lies in a spectral gap of $-\Delta+V$, then (*) has at least one nontrivial solution. If in addition $f$ is odd in $u$, then (*) has infinitely many (geometrically distinct) solutions. The proofs rely on a degree-theory and a linking-type argument developed in this paper.

[1]  L. Jeanjean.  Solutions in Spectral Gaps for a Nonlinear Equation of Schrödinger Type , 1994 .

[2]  V. Benci.  On critical point theory for indefinite functionals in the presence of symmetries , 1982 .

[3]  Vittorio Coti Zelati,et al.  Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials , 1991 .

[4]  Yanyan Li,et al.  On multibump' bound states for certain semilinear elliptic equations , 1992 .

[5]  Vieri Benci,et al.  Critical point theorems for indefinite functionals , 1979 .

[6]  M. Willem,et al.  Nontrivial solution of a semilinear Schrodinger equation , 1996 .

[7]  B. Buffoni,et al.  Existence of a nontrivial solution to a strongly indefinite semilinear equation , 1993 .

[8]  K Fan,et al.  Minimax Theorems. , 1953, Proceedings of the National Academy of Sciences of the United States of America.

[9]  P. Rabinowitz.  Minimax methods in critical point theory with applications to differential equations , 1986 .

[10]  Tassilo Küpper,et al.  Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation , 1992 .

[11]  Paul H. Rabinowitz,et al.  Homoclinic type solutions for a semilinear elliptic PDE on ℝn , 1992 .

[12]  P. Lions.  The concentration-compactness principle in the Calculus of Variations , 1984 .

[13]  P. Lions.  The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 , 1984 .

[14]  YanYan Li,et al.  Existence of solutions for semilinear elliptic equations with indefinite linear part , 1992 .