Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells

Abstract

In this paper, we consider the following Schrödinger equations with critical growthwhere N ≥ 4, 2 * is the critical Sobolev exponent, a(x) ≥ 0 and its zero sets are not empty, λ > 0 is a parameter, δ > 0 is a constant such that the operator (− + λa(x) − δ) might be indefinite for λ large. We prove that if the zero sets of a(x) have several isolated connected components 1 , · · · , k such that the interior of i (i = 1, 2, . . . , k) is not empty and ∂ i (i = 1, 2, . . . , k) is smooth. Then for λ sufficiently large, the equation admits, for any i ∈ {1, 2, · · · , k}, a solution which is trapped in a neighborhood of i . The key ingredients of the paper are using a flow argument and a combination of global linking and local linking.

On the critical Choquard equation with potential well

“…Suppose that the potential V (x) satisfies (1.2), (1.3) and the nonlinearity is of subcritical growth, the authors in [17] overcame the loss of compactness and applied the deformation flow arguments to build the multi-bump shaped solutions. Recently the existence of multi-bump shaped solutions for (1.9) with critical growth was also studied in [23,24,39], the main results there generalize and complement the theorems in [17]. We would also like to mention some related nonlocal problems in [26] and the references therein, there the existence of solutions of the nonlocal Schrödinger-Poisson system was investigated under the effect of critical growth assumption or potential well type function V (x).…”

“…However, if β > β 1 , the operator −∆ + λV (x) − β might be indefinite in H 1 (R N ). Moreover, the appearance of convolution type nonlinearities brings us a lot of difficulties and the techniques in [8,24,39] can not be applied to the Choquard equation directly. Thus, to look for solutions for equation (1.8), we need to develop new techniques to overcome the difficulties.…”

Existence of bound state solutions for p-Laplacian equation with critical growth

“…Moreover, for normalized solutions, we refer the reader to [5,41] and reference therein. For sign changing solutions, one can find some result in [19,24,42] and reference therein. In contrast to the case of ground state solution, normalized solutions and sign changing solutions, there are very few results for the existence of bound state solutions.…”

On critical Choquard equation with potential well

“…Suppose that the potential V (x) satisfies (2), (3) and the nonlinearity is of subcritical growth, the authors in [17] overcame the loss of compactness and applied the deformation flow arguments to build the multi-bump shaped solutions. Recently the existence of multi-bump shaped solutions for (9) with critical growth was also studied in [23,24,36], the main results there generalize and complement the theorems in [17]. The nonlocal Schrödinger-Poisson system was considered in [25], there the existence of solutions was investigated under the critical growth assumption or potential well type function V (x).…”

“…However, if β > β 1 , the operator −∆ + λV (x) − β might be indefinite in H 1 (R N ). Moreover, the appearance of convolution type nonlinearities brings us a lot of difficulties and the techniques in [8,24,36] can not be applied to the Choquard equation directly. Thus, to look for solutions for equation (8), we need to develop new techniques to overcome the difficulties.…”