Subcritical Nonlinearity Research Articles

On a cascade system of Schrödinger equations. Fractional powers approach

Our goal is to study fractional powers of a cascade system of partial differential equations. We explicitly calculate the fractional powers of linear operators associated to this type of system and we discuss local solvability of the fractional equation with subcritical nonlinearity. As an example, a cascade system of Schrödinger equation is analyzed. A connection between the fractional system and the original system is established and we prove the convergence of the linear semigroups obtained by the fractional power operator to the original linear semigroup, as the power α approaches 1.

Concentration of solutions for fractional double-phase problems: critical and supercritical cases

This paper is concerned with concentration and multiplicity properties of solutions to the following fractional problem with unbalanced growth and critical or supercritical reaction:{(−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=h(u)+|u|r−2u in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0, in RN,} where ε is a positive parameter, 0<s<1, 2⩽p<q<N/s, (−Δ)ts(t∈{p,q}) is the fractional t-Laplace operator, while V:RN↦R and h:R↦R are continuous functions. The analysis developed in this paper covers both critical and supercritical cases, that is, we assume that either r=qs⁎:=Nq/(N−sq) or r>qs⁎. The main results establish the existence of multiple positive solutions as well as related concentration properties. In the first case, due to the strong influence of the critical term, the result holds true for “high perturbations” of the subcritical nonlinearity. In the second framework, the result holds true for “low perturbations” of the supercritical nonlinearity. The concentration properties are achieved by combining topological and variational methods, provided that ε is small enough and in close relationship with the set where the potential V attains its minimum.

Existence and orbital stability of standing waves to a nonlinear Schr\xf6dinger equation with inverse square potential on the half-line

In our work, we establish the existence of standing waves to a nonlinear Schrödinger equation with inverse-square potential on the half-line. We apply a profile decomposition argument to overcome the difficulty arising from the non-compactness of the setting. We obtain convergent minimizing sequences by comparing the problem to the problem at “infinity” (i.e., the equation without inverse square potential). Finally, we establish orbital stability/instability of the standing wave solution for mass subcritical and supercritical nonlinearities respectively.

Normalized ground states for semilinear elliptic systems with critical and subcritical nonlinearities

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1u=\mu _1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ -\Delta v+\lambda _2v=\mu _2 |v|^{q-2}v+\beta r_2|u|^{r_1}|v|^{r_2-2}v\quad &{}\hbox {in}~{{\mathbb {R}}^N},\\ \int _{{{\mathbb {R}}^N}}u^2=a_1^2\quad \hbox {and}\quad \int _{{{\mathbb {R}}^N}}v^2=a_2^2, \end{array}\right. } \end{aligned}$$where \(p,r_1+r_2<2^*\) and \(q\le 2^*\). To this purpose, we study the geometry of the Pohozaev manifold and the associated minimization problem. Under some assumptions on \(a_1,a_2\) and \(\beta \), we obtain the existence of the positive normalized ground state solution to the above system.

Behavior Rigidity Near Non-Isolated Blow-up Points for the Semilinear Heat Equation

Abstract We consider the semilinear heat equation with Sobolev subcritical power nonlinearity in dimension $N=2$, and $u(x,t)$ a solution that blows up in finite time $T$. Given a non-isolated blow-up point $a$, we assume that the Taylor expansion of the solution near $(a,T)$ obeys some degenerate situation labeled by some even integer $m(a)\ge 4$. If we have a sequence $a_n \to a$ as $n\to \infty $, we show after a change of coordinates and the extraction of a subsequence that either ${a_{n,1}}-a_1 = o((a_{n,2}-a_2)^2)$ or $|a_{n,1}-a_1||a_{n,2}-a_2|^{-\beta } |\log |a_{n,2}-a_2||^{-\alpha } \to L&amp;gt; 0$ for some $L&amp;gt;0$, where $\alpha $ and $\beta $ enjoy a finite number of rational values with $\beta \in (0,2]$ and $L$ is a solution of a polynomial equation depending on the coefficients of the Taylor expansion of the solution. If $m(a)=4$, then $\alpha =0$ and either $\beta =3/2$ or $\beta =2$.

Ground states for fractional Schrödinger equations involving critical or supercritical exponent

In this paper, we study the following fractional Schrödinger equation involving critical or supercritical exponent where 0<s<1, N>2s, , , , denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for small by the Nehari method. Our main contribution is that we are able to deal with the supercritical case .

On Fractional Choquard Equation with Subcritical or Critical Nonlinearities

On Fractional Choquard Equation with Subcritical or Critical Nonlinearities

A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent

Let (M,g,σ) be a compact Riemannian spin manifold of dimension m≥2, let S(M) denote the spinor bundle on M, and let D be the Atiyah-Singer Dirac operator acting on spinors ψ:M→S(M). We study the existence of solutions of the nonlinear Dirac equation with critical exponent(NLD)Dψ=λψ+f(|ψ|)ψ+|ψ|2m−1ψ where λ∈R and f(|ψ|)ψ is a subcritical nonlinearity in the sense that f(s)=o(s2m−1) as s→∞. A model nonlinearity is f(s)=αsp−2 with 2<p<2mm−1, α∈R. In particular we study the nonlinear Dirac equation(BND)Dψ=λψ+|ψ|2m−1ψ,λ∈R. This equation is a spinorial analogue of the Brezis-Nirenberg problem. As corollary of our main results we obtain the existence of least energy solutions (λ,ψ) of (BND) for every λ>0, even if λ is an eigenvalue of D. For some classes of nonlinearities f we also obtain solutions of (NLD) for every λ∈R, except for non-positive eigenvalues. If m≢3 (mod 4) we obtain solutions of (BND) for every λ∈R, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it.The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD).

Bifurcation for nonlinear elliptic problems with singular potentials

We study the bifurcation phenomena for a class of nonlinear elliptic problems with Hardy term and subcritical nonlinearity. Sufficient conditions for bifurcation are established. The existence of a positive solution is also obtained for an asymptotically linear problem.

Hamiltonian systems of Schrödinger equations with vanishing potentials

In this paper, by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for a Hamiltonian system of Schrödinger equations in [Formula: see text] with potentials vanishing at infinity and subcritical nonlinearities which are superlinear at the origin and at infinity. We establish new estimates to prove the boundedness of a Cerami sequence.

Non-existence results for a degenerate semilinear elliptic equation

ABSTRACT Based upon the method of moving spheres, we establish some Liouville-type results of a weighted elliptic equation. For subcritical nonlinearity, we prove non-existence result on the entire space. For supercritical nonlinearity, we obtain non-existence on bounded star-shaped domains. Meanwhile, a regularity result is also obtained.

Sign-changing and constant-sign solutions for elliptic problems involving nonlocal integro-differential operators

This work is devoted to study the existence of sign-changing and constant-sign solutions for nonlinear elliptic problems driven by nonlocal integro-differential operators with subcritical nonlinearity which is not locally Lipschitz continuous and there is no (AR) condition and no any monotony properties. By using some variants of the Mountain Pass Lemma, we show the existence of a positive solution, a negative solution and a sign-changing solution.

Critical groups and multiple solutions for Kirchhoff type equations with critical exponents

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

Concentration phenomenon of solutions for a class of Kirchhoff-type equations with critical growth

In this paper, we study the following Kirchhoff-type equations with critical growth−(ε2a+εb∫R3|∇u|2dx)Δu+V(x)u=P(x)f(u)+Q(x)|u|4u,x∈R3, where ε>0, a>0, b>0 and f is a continuous superlinear but subcritical nonlinearity. Under suitable assumptions on the potentials V(x), P(x) and Q(x), we obtain the existence and concentration of positive solutions and prove that the semiclassical solutions wε with maximum points xε concentrating at a special set Sp characterized by V(x), P(x) and Q(x). Furthermore, for any sequence xε→x0∈Sp, vε(x):=wε(εx+xε) converges in H1(R3) to a ground state solution v of−(a+b∫R3|∇v|2dx)Δv+V(x0)v=P(x0)f(v)+Q(x0)|v|4v,x∈R3.

Nonhomogeneous systems involving critical or subcritical nonlinearities

This paper deals with existence of a nontrivial positive solution to systems of equations involving nontrivial nonhomogeneous terms and critical or subcritical nonlinearities. Via a minimization argument we prove existence of a positive solution whose energy is negative provided that the nonhomogeneous terms are small enough in the dual norm.

Existence of the non‐radially symmetric ground state for p‐Laplacian equations involving Choquard type

We consider some p‐Laplacian type equations with sum of nonlocal term and subcritical nonlinearities. We prove the existence of the ground states, which are positive. Because of including p=2, these results extend the results of Li, Ma and Zhang [Nonlinear Analysis: Real World Application 45(2019) 1‐25]. When p=2, N=3, by a variant variational identity and a constraint set, we can prove the existence of a non‐radially symmetric solution. Moreover, this solution u(x1, x2, x3) is radially symmetric with respect to (x1, x2) and odd with respect to x3.

Uniform bounds for higher-order semilinear problems in conformal dimension

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem (−Δ)mu=h(x,u) in Ω,u=∂nu=⋯=∂nm−1u=0 on ∂Ω,where h is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger–Moser–Adams inequality, either when Ω is a ball or, provided an energy control on solutions is prescribed, when Ω is a smooth bounded domain. Our results are sharp within the class of distributional solutions. The analogous problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory.

Nontrivial solutions for the fractional Schrödinger‐Poisson system with subcritical or critical nonlinearities

In this paper, we are concerned with a class of fractional Schrödinger‐Poisson system involving subcritical or critical nonlinearities. By using the Nehari manifold and variational methods, we obtain the existence and multiplicity of nontrivial solutions.

Existence and Profile of Ground-State Solutions to a 1-Laplacian Problem in $$\mathbb {R}^N$$

In this work we prove the existence of ground state solutions for the following class of problems $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle - \Delta _1 u + (1 + \lambda V(x))\frac{u}{|u|} &{} = f(u), \quad x \in \mathbb {R}^N, u \in BV(\mathbb {R}^N), &{} \end{array} \right. \end{aligned}$$ where $$\lambda > 0$$ , $$\Delta _1$$ denotes the 1-Laplacian operator which is formally defined by $$\Delta _1 u = \text{ div }(\nabla u/|\nabla u|)$$ , $$V:\mathbb {R}^N \rightarrow \mathbb {R}$$ is a potential satisfying some conditions and $$f:\mathbb {R} \rightarrow \mathbb {R}$$ is a subcritical nonlinearity. We prove that for $$\lambda > 0$$ large enough there exist ground-state solutions and, as $$\lambda \rightarrow +\infty $$ , such solutions converges to a ground-state solution of the limit problem in $$\Omega = \text{ int }( V^{-1}(\{0\}))$$ .

Positive Solutions for a Class of Quasilinear Schrödinger Equations with Two Parameters

In this paper, we consider the following quasilinear Schrodinger equation $$\begin{aligned} -\Delta u+V(x)u+\frac{\kappa }{2}\Delta (u^2)u=\lambda f(u)+h(u),\,\, x\in {\mathbb {R}}^N, \end{aligned}$$where $$\lambda ,\kappa >0$$, $$N\ge 3$$ and $$2^*=\frac{2N}{N-2}$$. By using a change of variable, we obtain the existence of positive solutions for this problem with subcritical nonlinearities by using the mountain pass theorem and Moser iterative method. Our results extend and supplement some other related literatures.