First and second critical exponents for an inhomogeneous Schrödinger equation with combined nonlinearities

We study the large-time behavior of solutions for the inhomogeneous nonlinear Schrödinger equation iut+Δu=λ|u|p+μ|∇u|q+w(x),t>0,x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} iu_t+\Delta u=\lambda |u|^p+\mu |\nabla u|^q+w(x),\quad t>0,\, x\in {\mathbb {R}}^N, \end{aligned}$$\end{document}where N≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 1$$\end{document}, p,q>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p,q>1$$\end{document}, λ,μ∈C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ,\mu \in {\mathbb {C}}$$\end{document}, λ≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \ne 0$$\end{document}, and u(0,·),w∈Lloc1(RN,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(0,\cdot ), w\in L^1_{\mathrm{loc}}({\mathbb {R}}^N,{\mathbb {C}})$$\end{document}. We consider both the cases where μ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =0$$\end{document} and μ≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \ne 0$$\end{document}, respectively. We establish existence/nonexistence of global weak solutions. In each studied case, we compute the critical exponents in the sense of Fujita, and Lee and Ni. When μ≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \ne 0$$\end{document}, we show that the nonlinearity |∇u|q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nabla u|^q$$\end{document} induces an interesting phenomenon of discontinuity of the Fujita critical exponent.


Introduction and main results
Let N ≥ 1, p, q > 1, λ, μ ∈ C with λ = 0. Our goal here is to investigate the existence and nonexistence of global weak solutions to the inhomogeneous nonlinear Schrödinger problem iu t + Δu = λ|u| p + μ|∇u| q + w(x) in(0, ∞) × R N , u(0, x) = u 0 (x) i n R N , (1.1) subject to the assumptions u 0 , w ∈ L 1 loc (R N , C). The nonlinearity in the right side of the main equation presents the combined effects of two potential terms, depending by powers of |u| and of |∇u|, respectively. Thus, when μ = 0 our problem is gradient dependent, and this requires some specific estimates in our arguments of proofs. However, we know that the dependence on gradient occurs in many physical models of (heat) transport and fluid mechanics of plasma. Thus, it is an important key feature to model "rich dynamics" in evolution processes. Here, the cases μ = 0 and μ = 0 will be discussed separately. Precisely, we will show the suitability of our approach for the gradient dependence (μ = 0) in the nonlinearity. We will develop this case as a perturbation of the starting case μ = 0. In both the situations, we will also point out the effects of the inhomogeneous term w ∈ L 1 loc (R N , C) on the global behavior of the problem. A crucial research direction is to establish the appropriate framework space, where the boundary data belong.
Starting with the case where μ = 0 and w ≡ 0, (1.1) reduces to the Cauchy problem Problem (1.2) has been studied by many authors, whose finding leads to the consideration of certain special exponents' values. For 1 < p < 1 + 4 N −2s , 0 ≤ s < N 2 , it is well known that local well-posedness for (1.2) holds in Sobolev spaces H s (R N ) (see, e.g., [1,17]). When N = 1 and p = 2, it was shown in [10] that (1.2) is locally well posed in H s (R N ), for s > − 1 4 . For arbitrary N , if u 0 ∈ L 2 (R N ) ∩ L 1+ 1 p (R N ) and is the Strauss exponent, the global existence for (1.2) for small initial data holds (see [16]). In [6], when 1 < p ≤ 1 + 2 N , it was shown that, if u 0 ∈ L 2 (R N ) and Re λ Im then the L 2 -norm of the solution to (1.2) blows up at finite time. Clearly, here by z ∈ C, Re z and Im z we mean the real part and the imaginary part of z, respectively. Later, in [4], when 1 < p < 1 + 4 N , a small initial data blow-up result of the L 2 -solution to (1.2) was derived. In [5], when p > 1 + 4 N , for suitable L 2 -data, it was shown that (1.2) admits no local weak solution. In [2], the authors extended the obtained results in [6] to the fractional Schrödinger equation where 0 < α < 2, u 0 ∈ H α 2 (R N ), and (−Δ) α 2 is the fractional Laplacian operator of order α 2 . Namely, they investigated the local well-posedness of solutions to (1.3) in H α 2 (R N ) and derived a finite-time blowup result, under suitable conditions on the initial data. In [11], the authors investigated the nonlocal in time nonlinear Schrödinger equation where 0 < α < 1 and J α 0|t is the Riemann-Liouville fractional integral operator of order α. Namely, they derived a blow-up exponent and obtained an estimate of the life span of blowing-up solutions to (1.4). In [19], the authors studied the time-fractional Schrödinger equation where 0 < α < 1 and ∂ α t is the Caputo fractional derivative of order α. Namely, it was shown that (1.5) admits no global weak solution with suitable initial data when 1 < p < 1 + 2 N . Moreover, the authors derived sufficient conditions for which (1.5) admits no global weak solution for every p > 1. For other works related to (1.2), see, e.g., [7][8][9]14] and the manuscripts cited therein. Motivated by the above-mentioned contributions, (1.1) is investigated in this paper. As already mentioned before, we first assume that μ = 0. Focusing on this case, we consider the problem Before stating in which sense solutions to (1.6) are considered, we indicate some of the notations used throughout this paper. Let Q = [0, ∞)×R N , we denote by C 2 c (Q, R) the space of C 2 real-valued functions compactly supported in Q. For z ∈ C, let z 1 = Re z and z 2 = Im z. Moreover, the symbol C will denote always a generic positive constant, which is independent of the scaling parameter T and the solution u.

Remark 1.2.
We underline the following two facts about the situations considered in Theorem 1.1: (i) The only condition on the initial value u 0 is that u 0 ∈ L 1 loc (R N , C). We just need this condition to guarantee that the integral term from the left side of (1.7) is well defined. (ii) For N ≥ 3, the critical exponent p * (N ) is the same as the one obtained by Zhang [18] for the inhomogeneous semilinear heat equation where w = w(x) ≥ 0, w ≡ 0. Namely, the following results were established in [18]: (b) If p > N N −2 , then problem (1.10) admits global positive solutions for some w, u 0 > 0. Notice that the proof of the nonexistence result in the case p = N N −2 makes use of Lemma 1 in [15], which holds only for u ≥ 0, while in our work, the solutions to (1.6) are complex-valued functions. The critical case p = N N −2 for problem (1.6) is left open. Next, we study the influence of the inhomogeneous term w on the critical behavior of (1.6) when N ≥ 3 and p > p * (N ) and determine the second critical exponent in the sense of Lee and Ni [12]. Just before, for σ < N, we introduce the following sets: We have the following bifurcation-type result.

Remark 1.3.
We underline the following two facts on Theorem 1.2: (i) Since p > N N −2 , we have that the set of σ satisfying σ * ≤ σ < N is nonempty. (ii) The second critical exponent for (1.6) in the sense of Lee and Ni is given by (1.11). Now, we consider (1.1) when μ = 0. Namely, our aim is to study the influence of the nonlinear term |∇u| q on the previous obtained results. Let us mention in which sense solutions to (1.1) are considered this time. Definition 1.2. Let λ, μ ∈ C, λ, μ = 0, p, q > 1, and u 0 , w ∈ L 1 loc (R N , C). We say that u is a global weak solution to (1.1), if the following conditions hold: (1.12) As previously, we first determine the critical exponent for (1.1) in the sense of Fujita.

Remark 1.5.
We point out the following two facts established in Theorem 1.3: (i) Similarly to the case μ = 0, the only condition on the initial value u 0 is that u 0 ∈ L 1 loc (R N , C). (ii) From Theorem 1.3, we deduce that for N ≥ 3, the critical exponent in the sense of Fujita for (1.1) is given by One observes that the gradient term induces an interesting phenomenon of discontinuity of the critical exponent, jumping from p = N N −2 (the Fujita critical exponent for (1.6)) to p = ∞ as q reaches the value N N −1 from above.
Next, for N ≥ 3, p > p * (N ), and q > q * (N ), we investigate the influence of the inhomogeneous term w on the critical behavior of (1.1).
(a) If σ < σ * , where σ * is defined by (1.11), and (ii) If N ≥ 3, p > p * (N ), and q > q * (N ), then the second critical exponent for (1.1) in the sense of Lee and Ni is given by The rest of the paper is organized as follows. In Sect. 2, we focus on the case μ = 0 and establish the proofs of Theorems 1.1 and 1.2. In Sect. 3, we extend the study to the case where μ = 0, and hence we establish the proofs of Theorems 1.3 and 1.4.

The case µ = 0
In this section, we prove Theorems 1.1 and 1.2, studying problem (1.6) (that is, problem (1.1) when μ = 0). We establish both the absence and the existence of global solutions in the sense of Definition 1.2. The proofs are based on a rescaled test function argument (see [13] for a general account of these methods).

Proof of Theorem 1.1
is a global weak solution to (1.6). We first consider the case Since the second assumption in (2.1) holds, it is obvious that λ 1 = 0. So, using the integral equation for all ϕ ∈ C 2 c (Q, R) with ϕ ≥ 0 and ϕ(0, ·) ≡ 0. This yields the following inequality in u With respect to the right side of (2.3), the ε-Young inequality for 0 < ε < 1 2 gives us two estimates: and Turning to (2.3), and using the estimates (2.4) and (2.5) in the right side, we deduce that which yields At this step, we consider two cutoff functions ξ, η ∈ C ∞ ([0, ∞)) satisfying, respectively, For sufficiently large T , we define the following function: where k > 2p p−1 and ρ > 0 are constants. We observe that ϕ T ∈ C 2 c (Q, R), ϕ T ≥ 0, and ϕ T (0, ·) ≡ 0. So, for sufficiently large T , by (2.7) and using elementary calculations, we obtain the following: Hence, by (2.7) and (2.8), for sufficiently large T we obtain Similarly, using (2.7) and (2.9), for sufficiently large T we obtain On the other hand, Since w 1 ∈ L 1 (R N , R) by (2.1), the dominated convergence theorem leads to Again by (2.1), we know that and hence for sufficiently large T , we get Hence, it follows from (2.12) and (2.13) that (2.14) Therefore, for sufficiently large T , taking ϕ = ϕ T in (2.6) and using (2.10), (2.11), and (2.14), we obtain Observe that for ρ = 1 2 , we have So, taking ρ = 1 2 in (2.15), we obtain 1 λ 1 Since 1 < p < p * (N ), passing to the limit as T → ∞ in (2.16), we arrive to a contradiction to Consider now the case Taking u 1 = u 2 , u 2 = −u 1 , u 01 = u 02 , u 02 = −u 01 , u = u 1 + i u 2 , λ 1 = λ 2 , and w 1 = w 2 , we deduce from (1.9) and (2.17) that for every ϕ ∈ C 2 c (Q, R). Hence, from the previous case, we obtain a contradiction with λ 1 Therefore, in both cases (case (2.1) and case (2.17)), (1.6) admits no global weak solution. This proves part (i) of Theorem 1.1.
(ii) To construct the second part of the proof, we focus on the situation where We consider the class of functions Using the constraints (2.20) and (2.21), we obtain the following sign information for w:

Proof of Theorem 1.2
This proof is based on the previous one, but it is slightly modified according to the different assumptions.
Since some a priori estimates used in the proof of Theorem 1.1 still work, we remain to consider certain additional factors as follows.
(i) Suppose that u ∈ L p loc (Q, C) is a global weak solution to (1.6). Without restriction of the generality, we suppose that (2.23) The case λ 2 w 2 ∈ Λ σ can be treated in a similar way (see the case (2.17) in the proof of part (i) of Theorem 1.1).
For sufficiently large T , we have where ϕ T is again defined by (2.7). Hence, by the definition of the function η, and by (2.23), for sufficiently large T , we obtain Hence, taking ϕ = ϕ T in (2.6) and using (2.10), (2.11), and (2.24), we obtain that is, Taking ρ = 1 2 in the above inequality, we get Since σ < σ * = 2p p−1 , passing to the limit as T → ∞ in (2.25), we obtain a contradiction with C > 0. This proves part (i) of Theorem 1.2.
(ii) In the second part of the proof, we focus on the case where We consider the class of functions (2.19), with and ε satisfying (2.21). Notice that since σ < N, the set of values of δ satisfying (2.26) is nonempty. Moreover, since σ * ≤ σ, we have Hence, by (2.26), we deduce that δ satisfies (2.20). Therefore, from the proof of part (ii) of Theorem 1.1, for all δ and ε satisfying, respectively, (2.26) and (2.21), the function u δ,ε defined by (2.19) is a stationary positive solution to (1.6), where λ = −1 and w = w(x) (< 0) is defined by (2.22). Now, we have just to show that λw ∈ Σ σ , that is, Hence, using (2.26), for sufficiently large r, we obtain and hence (2.27) is established. This fact proves part (ii) of Theorem 1.2.
(ii) To construct the second part of the proof, we focus on the situation where Consider the class of functions u δ,ε defined by (2.19), where and (3.14) Notice that due to (3.12), the set of δ satisfying (3.13) is nonempty. Let

Proof of Theorem 1.4
(i) Let N ≥ 3, p > N N −2 , and q > N N −1 . Suppose that u is a global weak solution to (1.1). We first consider the case for some i ∈ {1, 2}. Without restriction of the generality, we may suppose that i = 1, that is, From (2.24), for sufficiently large T , we have Next, following the same arguments used in the proof of the statement (i) of Theorem 1.2, we obtain (2.25), which leads to a contradiction with C > 0. This proves part (i)-(a) of Theorem 1.4. Consider now the case for some i ∈ {1, 2}. Without restriction of the generality, we may suppose that i = 1, that is, Similar to the previous case, for sufficiently large T , we obtain Hence, taking ρ = p(q−1) (p−1)q and using (3.9), for sufficiently large T , it holds that (3.16) Since σ < σ * * , passing to the limit as T → ∞ in (3.16), we arrive to contradiction to C > 0. This proves part (i)-(b) of Theorem 1.4.
From (3.18), for sufficiently large |x|, we obtain This shows that λw ∈ Σ σ , and so also part (ii) of Theorem 1.4 is established.

Conclusions
This paper aimed to enlarge the discussion about the existence and nonexistence of solutions to certain Schrödinger equations, with suitable nonlinearities. We centered on global weak solutions in the sense of Definitions 1.1 and 1.2. So, we studied first the situation where we drop the |∇u| q -dependence of the nonlinearity (case μ = 0), and then we established the similar results in the situation where the nonlinearity depends by |∇u| q (case μ = 0). The second case can be seen as a way to perform a perturbation analysis of the response of the Schrödinger equation to changes in nonlinearity properties. In particular, we observed that the gradient term induced a phenomenon of discontinuity of the critical exponent, jumping from the Fujita critical exponent for (1.6) (p = N N −2 ) to the value p = ∞ as q reaches the value N N −1 from the above (see Theorem 1.3). The use of some classical rescaled test function arguments linked to the Mitidieri-Pokhozhaev method [13] was the crucial key in establishing the proofs, in a clear way.