Existence of multi-bump solutions for nonlinear Schroedinger equations via variational method
- McMaster Univ., Hamilton, Ontario (Canada)
In this paper, we consider the general semilinear equation (1.6) {epsilon}{sup 2}{Delta}{sub u}-V(x)u+f(u)=O, x {epsilon} R{sup N}. We assume that V(x) satisfies (V1) V(x) is locally Hoelder continuous in R{sup N} and V(x){ge}V{sub O}>O; (V2) there exist k disjoint bounded regions {Omega}{sub 1}, {Omega}{sub 2}...,{Omega}{sub k} such that (1.7) M{sub i}:=inf/{partial_derivative}{Omega}{sub i}/V(x) > {omega}{sub i}: = inf/{Omega}{sub i}V(x), i=1,2,...,k. We also assume that f(u) satisfies (f1) f(u) {epsilon} C{sup 1}(R), f(u){equivalent_to}O for u {le} O and f(u){equivalent_to} O for u > O; (f2) f(u)/u is nondecreasing in u; (f3) O {le} f{sub u}(u) {le} a{sub 1} + a{sub 2}u{sup p-1} for some positive constants a{sub 1}, a{sub 2} and 1 < p < N+2/N-2 (we use the convention now and later that N+2/N-2 should be replaced by {infinity} when N = 1,2); (f4) there exists {beta} {epsilon} (O,1/2) such that F(u) {le} {beta}uf(u), u{ge}O where F(u) = {integral}{sub O}{sup u} f(t) dt. This paper is organized as follows. In Section 2, we scale the equation (1.6) properly and present some results for generalized Palais Smale sequences of a family of modified functionals. In Section 3, we show the existence of critical points of the modified functionals at certain energy level and in certain neighborhood. Finally, we show the concentration property of these critical points and therefore obtain the solution for equation (1.6). This also finishes the proof of the main theorem. I have learned from the referee that he has just received a closely related paper by del Pino and Felmer, but which is based on different methods. 27 refs.
- Sponsoring Organization:
- USDOE
- OSTI ID:
- 437120
- Journal Information:
- Communications in Partial Differential Equations, Vol. 21, Issue 5-6; Other Information: PBD: 1996
- Country of Publication:
- United States
- Language:
- English
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