Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

Abstract

In this paper, we consider the following nonlinear Schrödinger equations with mixed nonlinearities:where N ≥ 3, µ > 0, λ ∈ R and 2 < q < 2 * . We prove in this paper (1) Existence of solutions of mountain-pass type for N = 3 and 2 < q < 2 + 4 N .(2)Existence and nonexistence of ground states for 2 + 4 N ≤ q < 2 * with µ > 0 large.(3)Precisely asymptotic behaviors of ground states and mountain-pass solutions as µ → 0 and µ goes to its upper bound. Our studies answer some questions proposed by Soave in [49, Remarks 1.1, 1.2 and 8.1].

On some nonlinear Schrödinger equations in $\bbr^N$

“…Remark 1.2. Theorem 1.3, together with our recent study in [31], gives a completed answer to the open question proposed by Soave in [30].…”

“…for all r ≥ 0 uniformly as n → ∞. Now, we can adapt the ODE's argument in [5,17,19] as that in the proof of [31,Lemma 4.1] to obtain…”

“…Thus, by our above observations, normalized solutions of (1.4) is equivalent to fixed-frequency solutions of (1.4) with another additional condition. Since we make a detail study on some special normalized solutions of (1.4) in [31], we could use these detail estimates to derive Theorem 1.2.…”

“…On the other hand, since m(t) is the minimum of E t (v) on the Nehari manifold N t , it is standard (cf. [31,Lemma 3.3]) to use the fibering maps (3.1) to show that m(t) is nonincreasing for t > 0. Note that it is well known that m(0…”

“…Based on our recent study on the normalized solutions of the above equation in [31], we prove that (1) the above equation has two positive radial solutions for N = 3, 2 < q < 4 and t > 0 sufficiently large, which gives a rigorous proof of the numerical conjecture in [14]; (2) there exists t * q > 0 for 2 < q ≤ 4 such that the above equation has ground-states for t ≥ t * q in the case of 2 < q < 4 and for t > t * 4 in the case of q = 4 while, the above equation has no ground-states for 0 < t < t * q for all 2 < q ≤ 4, which, together with the well-known results on groundstates of the above equation, almost completely solve the existence of ground-states to the above equation, except for N = 3, q = 4 and t = t * 4 . Moreover, based on the almost completed study on ground-states to the above equation, we introduce a new argument to study the normalized solutions of the above equation to prove that there exists 0 < ta,q < +∞ for 2 < q < 2+ 4 N such that the above equation has no positive normalized solutions for t > ta,q with R N |u| 2 dx = a 2 , which, together with our recent study in [31], gives a completed answer to the open question proposed by Soave in [30]. Finally, as applications of our new argument, we also study the following Schrödinger equation with a partial confinement:…”