Nonexistence and existence of positive radial solutions to a class of quasilinear Schrödinger equations in RN$\mathbb{R}^{N}$

<jats:p>This paper aims to investigate the class of quasilinear Schrödinger equations <jats:disp-formula><jats:alternatives><jats:tex-math> $$\begin{aligned} \begin{aligned}[b] &-\Delta u-\bigl[\Delta \bigl(1+u^{2}\bigr)^{\frac{\gamma }{2}}\bigr] \frac{\gamma u}{2(1+u^{2})^{\frac{2-\gamma }{2}}}\\ &\quad =\alpha h\bigl( \vert x \vert \bigr) \vert u \vert ^{p-1}u+ \beta H\bigl( \vert x \vert \bigr) \vert u \vert ^{q-1}u, \quad x\in \mathbb{R}^{N}, \end{aligned} \end{aligned}$$ </jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mtable><mml:mtr><mml:mtd /><mml:mtd><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mi>u</mml:mi><mml:mo>−</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mi>Δ</mml:mi><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mfrac><mml:mi>γ</mml:mi><mml:mn>2</mml:mn></mml:mfrac></mml:msup><mml:mo>]</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>γ</mml:mi><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:msup><mml:mi>u</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>γ</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:msup></mml:mrow></mml:mfrac></mml:mtd></mml:mtr><mml:mtr><mml:mtd /><mml:mtd><mml:mspace /><mml:mo>=</mml:mo><mml:mi>α</mml:mi><mml:mi>h</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>|</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:msup><mml:mo>|</mml:mo><mml:mrow><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:mi>H</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>|</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:mo>)</mml:mo></mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:msup><mml:mo>|</mml:mo><mml:mrow><mml:mi>q</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mspace /><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mi>N</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></jats:alternatives></jats:disp-formula> where <jats:inline-formula><jats:alternatives><jats:tex-math>$N >2$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>></mml:mo><mml:mn>2</mml:mn></mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$1 \le \gamma \le 2$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>γ</mml:mi><mml:mo>≤</mml:mo><mml:mn>2</mml:mn></mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$\alpha ,\beta \in \mathbb{R}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>∈</mml:mo><mml:mi>R</mml:mi></mml:math></jats:alternatives></jats:inline-formula> and either <jats:inline-formula><jats:alternatives><jats:tex-math>$0< p<1<q$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>q</mml:mi></mml:math></jats:alternatives></jats:inline-formula> or <jats:inline-formula><jats:alternatives><jats:tex-math>$1< p< q$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>q</mml:mi></mml:math></jats:alternatives></jats:inline-formula>. Functions <jats:inline-formula><jats:alternatives><jats:tex-math>$h(|x|)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>h</mml:mi><mml:mo>(</mml:mo><mml:mo>|</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$H(|x|)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mo>|</mml:mo><mml:mi>x</mml:mi><mml:mo>|</mml:mo><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula> are continuous and positive in <jats:inline-formula><jats:alternatives><jats:tex-math>$\mathbb{R}^{N} $</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>R</mml:mi><mml:mi>N</mml:mi></mml:msup></mml:math></jats:alternatives></jats:inline-formula>. Relying on some special arguments and the Schauder–Tychonoff fixed point theorem, nonexistence criteria, existence of positive ground state solutions and blow-up solutions to Eq. (0.1) with <jats:inline-formula><jats:alternatives><jats:tex-math>$0< p<1<q$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>q</mml:mi></mml:math></jats:alternatives></jats:inline-formula> or <jats:inline-formula><jats:alternatives><jats:tex-math>$1< p< q$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mi>q</mml:mi></mml:math></jats:alternatives></jats:inline-formula> will be obtained.</jats:p>


Preliminaries
This paper is concerned with the following quasilinear Schrödinger equation: 2-γ 2 = αh |x| |u| p-1 u + βH |x| |u| q-1 u, x ∈ R N , (1.1) where 1 ≤ γ ≤ 2, α, β ∈ R and either 0 < p < 1 < q or 1 < p < q. This class of equations is often referred to as so-called modified nonlinear Schödinger equations due to the quasilinear term [ (1 + u 2 ) , whose solutions are related to the standing wave solutions for the quasilinear Schrödinger equation where V is a given potential, Ψ and h are real functions.
Besides, Zhang, Liu, Wu and Cui [12] focused on the existence and nonexistence of entire blow-up solutions for the following quasilinear p-Laplacian Schrödinger equation with a non-square diffusion term: where p ≥ 2γ , γ > 1 2 , the nonnegative radial function q is continuous on R N , g is a continuous positive and non-decreasing function on [0, ∞). Chen and Chen [13] concentrated on the nonexistence of stable solutions for the quasilinear Schrödinger equation 5) where N ≥ 3, q > 5 2 , h(x) is continuous and positive in R N . Throughout the paper, we consider (1.1) with the following two cases: With the aid of a variational argument, the question of the existence and multiplicity of nontrivial solutions to problem (1.1) is largely open. Compared with the work on weak solutions by variational way, we are interested in investigating the radial solutions and asymptotic behavior. In the present paper, the first task is to obtain the nonexistence criteria of positive ground state solutions to (1.1) involving superlinear nonlinearities, which mainly relies on some special techniques. Immediately after that, the sufficient conditions Motivated by [14][15][16][17], we take the changing of variables u = g(z) or z = g -1 (u), where g(t) is given by and g(t) = -g(-t) on (-∞, 0]. Thus, we can obtain the properties of the function g(t) as below.
The function g(t) satisfies (f 1 ) g is uniquely defined, C ∞ and invertible; After making the change u = g(z), (1.1) turns into the following equation: where and in the sequel, We observe that z = z(|x|) = z(r) is a positive radial solution of (1.6) if and only if the function z(r) satisfies the following equation: As usual, we focus on the existence and nonexistence of weak solutions to (1.1) via (1.7). Our main conclusions in this work are as below.
, for all t > 0, then problem (1.7) has at least one positive ground state solution.
The organization of this work is as below. Sufficient conditions for nonexistence of positive ground state solutions to (1.7) will be set up in Sect. 2. Section 3 and Sect. 4 contain the proof of the existence of positive ground state solutions and blow-up solutions.

Nonexistence criteria of positive ground state solutions
In this section, we aim at deriving some useful lemmas by special techniques and then finishing the proof of Theorem 1.1. Throughout the paper, a function z is called a ground state solution of problem (1.1) if the weak solution z tends to zero as |x| → ∞.
In fact, we have M(t) > 0 for every t > 0. Otherwise, there exists t 0 > 0 such that M(t 0 ) < 0, then Integrating the above inequality over [t 0 , r], one can see which gives rise to a contradiction with the positive solution z(r). Therefore, M(r) = (N -2)z + rz = r N-2 z r 3-N > 0, for all r > 0.
Proof of Theorem 1.1 By (2.2), one can get Since z(r) is a positive ground state solution and g(z) satisfies properties Note that r N-2 z(r) is an increasing function, then where b = min{b 1 , b 2 }. By (P 1 ), A 1 (r) → ∞ or A 2 (r) → ∞ as r → ∞, it gives rise to a contradiction. Thus, there is no positive ground state solution to (1.7). Otherwise, A 1 (r) < ∞ and A 2 (r) < ∞ as r → ∞. Denote which implies that B 1 (s), B 2 (s) are bounded for all s > 0. One can see that On the other hand, one can have and by (2.5), one can obtain  (2.8) which implies that Since the function F(s) is increasing, we have F(s) ≤ 0. It yields a contradiction.
As to the other case, if min{F p (s), F q (s)} = F p (s), we can apply the same argument. Thus, Eq. (1.7) has no positive ground state solution and the proof is completed.

Remark 2.1 Since
we can get
and the operator T : It is easy to get T X ⊂ X, the operator T is continuous and relatively compact. Therefore there exists a z ∈ X such that T z = z holds by the Schauder-Tychonoff fixed point theorem.

Existence of positive blow-up solutions
In this section, we investigate (1.7) with concave-convex nonlinearities and give a proof of Theorem 1.3. Throughout the paper, a function z is called a blow-up solution of problem (1.1) if a weak solution z satisfies z → ∞ as |x| → ∞.