Ground State Solutions Research Articles

Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.

Ground state sign-changing solutions for Schrödinger–Kirchhoff-type problem with critical growth

In this paper, we study the following Schrödinger–Kirchhoff-type problem: −a+b∫R3|∇u|2dx△u+u=|u|4u+f(u), x∈R3, where a > 0 and b > 0 are small enough. Under suitable assumptions on f, we obtain the existence of ground state sign-changing solution ub by the constraint variational method with Miranda’s theorem. Moreover, we prove that its energy is strictly larger than twice that of the ground state solution.

Ground state solutions for Schrödinger–Poisson system with critical growth and nonperiodic potential

This paper is concerned with the existence of ground state solutions for the Schrödinger–Poisson system −Δu + V(x)u + ϕu = |u|4u + λ|u|p−2u in R3 and −Δϕ = u2 in R3, where λ > 0 and p ∈ [4, 6). Here, V(x)∈C(R3,R), V(x) = V1(x) for x1 > 0, and V(x) = V2(x) for x1 < 0, where V1, V2 are periodic functions in each coordinate direction. In this paper, we give a splitting lemma corresponding to the nonperiodic potential and, then, prove the existence of ground state solutions for any λ > 0 when p ∈ (4, 6). Moreover, when p = 4, the above system possesses a ground state solution for λ > 0 sufficiently large. It is worth underlining that the technique employed in this paper is also valid for the Sobolev subcritical problem studied by Cheng and Wang [Discrete Contin. Dyn. Syst., Ser. B 27, 6295 (2022)].

Hybrid Annealing Method Based on subQUBO Model Extraction With Multiple Solution Instances

Ising machines are expected to solve combinatorial optimization problems efficiently by representing them as Ising models or equivalent quadratic unconstrained binary optimization (QUBO) models . However, upper bound exists on the computable problem size due to the hardware limitations of Ising machines. This paper propose a new hybrid annealing method based on partial QUBO extraction, called subQUBO model extraction, with multiple solution instances. For a given QUBO model, the proposed method obtains <inline-formula><tex-math notation="LaTeX">$N_I$</tex-math></inline-formula> quasi-optimal solutions (quasi-ground-state solutions) in some way using a classical computer. The solutions giving these quasi-optimal solutions are called <i>solution instances</i> . We extract a size-limited subQUBO model as follows based on a strong theoretical background: we randomly select <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">$(N_S&lt;N_I)$</tex-math></inline-formula> solution instances among them and focus on a particular binary variable <inline-formula><tex-math notation="LaTeX">$x_i$</tex-math></inline-formula> in the <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math></inline-formula> solution instances. If <inline-formula><tex-math notation="LaTeX">$x_i$</tex-math></inline-formula> value is much <i>varied</i> over <inline-formula><tex-math notation="LaTeX">$N_S$</tex-math></inline-formula> solution instances, it is included in the subQUBO model; otherwise, it is not. We find a (quasi-)ground-state solution of the extracted subQUBO model using an Ising machine and add it as a new solution instance. By repeating this process, we can finally obtain a (quasi-)ground-state solution of the original QUBO model. Experimental evaluations confirm that the proposed method can obtain better quasi-ground-state solution than existing methods for large-sized QUBO models.

Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

Ground State Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

Ground State Solution of Kirchhoff Problems with Hartree Type Nonlinearity

Ground State Solution of Kirchhoff Problems with Hartree Type Nonlinearity

Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝ N

In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ 2 u − a + b ∫ ℝ N ∇ u 2 d x Δ u + V x u = I α ∗ F u f u + λ u 2 ∗ ∗ − 2 u , in ℝ N , u ∈ H 2 ℝ N , where a &gt; 0 , b ≥ 0 , λ is a positive parameter, α ∈ N − 2 , N , 5 ≤ N ≤ 8 , V : ℝ N ⟶ ℝ is a potential function, and I α is a Riesz potential of order α . Here, 2 ∗ ∗ = 2 N / n − 4 with N ≥ 5 is the Sobolev critical exponent, and Δ 2 u = Δ Δ u is the biharmonic operator. Under certain assumptions on V x and f u , we prove that the equation has ground state solutions by variational methods.

Ground state solutions and periodic solutions with minimal periods to second-order Hamiltonian systems

In this paper, we study the existence of periodic solutions to second-order Hamiltonian systems. The nonlinear term has a special form, which satisfies a nondecreasing monotone assumption. Making use of a generalized Nehari manifold, we built the homomorphism between the Nehari manifold and the unit sphere of some space. Some sufficient conditions are obtained to guarantee the existence of periodic solutions with the prescribed minimal period to Hamiltonian systems. Our results extend the results in references.

Polyharmonic Kirchhoff-type problem without Ambrosetti–Rabinowitz condition

In this paper, we will study the polyharmonic Kirchhoff-type problem with singular exponential nonlinearity without the Ambrosetti–Rabinowitz condition: { − M ( ∫ Ω | ∇ m u | n m ) Δ n m m u = f ( x , u ) | x | σ in Ω , u = ∇ u = ⋯ = ∇ m − 1 u = 0 on ∂ Ω , where Ω ⊂ R n is a bounded domain with smooth boundary, 0 < σ < n , n ≥ 2 m ≥ 2 , M is a Kirchhoff function and f ( x , u ) has critical exponential growth. We use a suitable version of the Mountain Pass Theorem to prove the existence of a positive ground state solution for this problem.

Existence and concentration of ground state solutions for an equation with steep potential well and exponential critical growth

In this paper we study the existence and concentration of ground state solution for the following class of elliptic equations{−div(a(|∇u|p)|∇u|p−2∇u)+(1+μV(x))b(|u|p)|u|p−2u=f(u)in RN,u∈W1,p(RN)∩W1,N(RN), where 1<p≤N, N≥2, μ>0 is a parameter, the functions a,b∈C1(R) and the nonlinearity f has an exponential critical growth. To show the existence and concentration results we obtain interesting estimates which do not depend on μ.

Ground state solution for the Schrödinger equation with Hardy potential and critical Sobolev exponent

In this paper, we consider the following Schrödinger equation (0.1) − Δ u − μ u | x | 2 + V ( x ) u = K ( x ) | u | 2 ∗ − 2 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ⩾ 4, 0 ⩽ μ &lt; μ ‾, μ ‾ = ( N − 2 ) 2 4 , V is periodic in x, K and f are asymptotically periodic in x, we take advantage of the generalized Nehari manifold approach developed by Szulkin and Weth to look for the ground state solution of (0.1).

Fractional Sobolev embedding with radial potential

We present some embedding results of fractional Ds,2(RN) space with radial potential. To overcome the difficulty of non-compactness, we obtain the Lions-type theorems by means of the refined Sobolev inequality with the Morrey norm. Then we apply the obtained results to study the existence of ground state solution for fractional equations with radial potential.

Ground state solution for critical Schrödinger equation with harmonic potential

We consider the existence and regularity of solution to the critical Schrödinger equation with harmonic potential. We generalize an earlier theorem due to Berestycki and Lions (1983) [3] about the subcritical case to the critical case.

Ground state solution for a nonlinear fractional magnetic Schrödinger equation with indefinite potential

This paper is concerned with the following nonlinear fractional Schrödinger equation with a magnetic field: ε2s(−Δ)A/εsu+V(x)u=f(|u|2)u inRN, where ɛ &amp;gt; 0 is a parameter, s ∈ (0, 1), N ≥ 3, V:RN→R and A:RN→RN are continuous potentials, and V may be sign-changing; the nonlinearity is superlinear with subcritical growth but without satisfying the Ambrosetti–Rabinowitz condition. Based on the Nehari manifold method, concentration-compactness principle, and variational methods, we prove the existence of a ground state solution for the above equation when ɛ is sufficiently small. Our results improve and extend the result of Ambrosio and d’Avenia [J. Differ. Equations 264, 3336–3368 (2018)].

The Choquard logarithmic equation involving a nonlinearity with exponential growth

In the present work we are concerned with the Choquard Logarithmic equation $-\Delta u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u)$ in $\mathbb{R}^2$, for $ a>0 $, $ \lambda >0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.

Ground state solutions for critical Schrödinger equations with Hardy potential

In this paper, we investigate the following Schrödinger equation 0.1 −Δu−μ|x|2u=g(u)+|u|2*−2uinRN\\{0}, where N ⩾ 3, , is called the critical Sobolev exponent and g satisfies some appropriate subcritical conditions. For any , we prove that problem (0.1) has a positive radial ground state solution, which possesses exponential decaying property at infinity and blow-up property at origin. Moreover, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state solutions to problem (0.1) converges to a ground state solution of 0.2 −Δu=g(u)+|u|2*−2uinRN. when μ < 0, we prove that the mountain pass level of problem (0.1) in cannot be achieved. Further, we obtain a ground state radial solution of problem (0.1) whose energy is strictly greater than the mountain pass level in . Also, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state radial solutions to problem (0.1) converges to a ground state radial solution of the limiting problem as n → ∞.

Semiclassical Ground State Solutions for a Class of Kirchhoff-Type Problem with Convolution Nonlinearity

Semiclassical Ground State Solutions for a Class of Kirchhoff-Type Problem with Convolution Nonlinearity

Ground state solutions for fractional p-Kirchhoff equation

We study the fractional p-Kirchhoff equation $$ \Big( a+b \int_{\mathbb{R}^N}{\int_{\mathbb{R}^N}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dx\, dy\Big) (-\Delta)_p^s u-\mu|u|^{p-2}u=|u|^{q-2}u, \quad x\in\mathbb{R}^N, $$ where \((-\Delta)_p^s\) is the fractional p-Laplacian operator, a and b are strictly positive real numbers, \(s \in (0,1)\), \(1 &lt; p&lt; N/s,\) and \(p&lt; q&lt; p^*_s-2\) with \(p^*_s=\frac{Np}{N-ps}\). By using the variational method, we prove the existence and uniqueness of global minimum or mountain pass type critical points on the \(L^p\)-normalized manifold\(S(c):=\big\{u\in W^{s,p}(\mathbb{R}^N): \int_{\mathbb{R}^N} |u|^pdx=c^p\big\}\).

Cylindrical Solutions and Ground State Solutions to Weighted Kirchhoff Equations

Cylindrical Solutions and Ground State Solutions to Weighted Kirchhoff Equations

Normalized ground states of nonlinear biharmonic Schrödinger equations with Sobolev critical growth and combined nonlinearities

This paper is devoted to studying the following nonlinear biharmonic Schrödinger equation with combined power-type nonlinearities Δ2u−λu=μ|u|q−2u+|u|4∗−2uinRN,where N≥5, μ>0, 2<q<2+8N, 4∗=2NN−4 is the H2-critical Sobolev exponent, and λ appears as a Lagrange multiplier. By analyzing the behavior of the ground state energy with respect to the prescribed mass, we establish the existence of normalized ground state solutions. Furthermore, all ground states are proved to be local minima of the associated energy functional.