Existence of multi-bumb solutions for nonlinear schrödinger equations via variational method
Existence of multi-bumb solutions for nonlinear schrödinger equations via variational method
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DOI：
10.1080/03605309608821208
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发表时间：
1996-12
影响因子：
1.9
通讯作者：
C. Gui
中科院分区：
文献类型：
--
作者：
C. Gui
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 levelmore » 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.« less
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 levelmore » 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.« less