Nonexistence Of Ground State Solutions Research Articles

Positive solutions for Kirchhoff equation in exterior domains with small Sobolev critical perturbation

ABSTRACT The paper concerns with the existence of positive solutions for the following Kirchhoff problem { − ( a + b ∫ Ω | ∇u | 2 d x ) Δu + u = | u | p − 2 u + ε | u | 4 u in Ω , u ∈ H 0 1 ( Ω ) , where a, b>0, 4<p<6, 0 $ ]]> ε > 0 is a parameter and Ω ⊂ R 3 is an exterior domain, that is, Ω is an unbounded domain with R 3 ∖Ω nonempty and bounded. After showing the nonexistence of ground state solutions, we prove the existence of one positive solution with higher energy when R 3 ∖Ω is contained in a small ball and 0 $ ]]> ε > 0 is sufficiently small. The novelty of this paper is that we extend the result obtained in Alves and de Freitas [Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Milan J Math. 2017;85:309–330] to nonlocal case and generalize the subcritical nonlinearity discussed in Chen and Liu [Positive solutions for Kirchhoff equation in exterior domains. J Math Phys. 2021;62:Article ID 041510] to small critical perturbation.

Existence and nonexistence of ground state solutions for a Schrödinger–Poisson–Slater equation

In this article, we are concerned with the Schrödinger–Poisson–Slater equation where , , p>1, q>1, . By using the Nehari–Pohozaev manifold, we obtain the existence and nonexistence of ground state solutions results, depending on the parameters p and q. Our results extends some recent work in the literature.

Existence and concentration of ground state solutions for alogarithmic Schr&amp;ouml;dinger equation with the critical exponent

We study a class of logarithmic Schrödinger equations involving competing for weight potentials and critical nonlinearities on an unbounded domain. The corresponding energy functional has no continuous first derivative in its natural Sobolev space. Firstly, using the non-smooth critical point theory, we show that the 问题 admits one ground state positive 解. Then, we relate the concentration of solutions to a global minimum set of a suitable ground energy function via a Nehari manifold method. Finally, we give a sufficient condition forthe nonexistence of ground state solutions.

Nonexistence of ground state solutions for generalized quasilinear Schrödinger equations via dual approach

We study quasilinear Schrödinger equations of the form −divA(u)∇u+12A′(u)|∇u|2+V(x)u=h(u), x∈RN, where N≥3,A∈C1(R,R) is a positive function, V∈C2(RN,R) is a given potential, and h∈C1(R,R) is a suitable nonlinearity. Under some mild assumptions, we establish the nonexistence of ground state solutions for such equations by using the dual variational approach and Pohožaev manifold technique.

Periodic and asymptotically periodic quasilinear elliptic systems

In this work, we are concerned with the existence and nonexistence of ground state solutions for the following class of quasilinear Schrödinger coupled systems taking into account periodic or asymptotically periodic potentials. The nonlinear terms are superlinear at infinity and at the origin. By using a change of variable, we turn the quasilinear system into a nonlinear system where we can establish a variational approach with a fine analysis on the Nehari method. For the nonexistence result, we compare the potentials with periodic potentials proving the nonexistence of ground state solutions.

Positive Solutions and Infinitely Many Solutions for a Weakly Coupled System

We study a Schrodinger system with the sum of linear and nonlinear couplings. Applying index theory, we obtain infinitely many solutions for the system with periodic potentials. Moreover, by using the concentration compactness method, we prove the existence and nonexistence of ground state solutions for the system with close-to-periodic potentials.

Positive solutions for a Kirchhoff-type problem involving multiple competitive potentials and critical Sobolev exponent

In this paper, we study the existence and multiplicity of positive solutions for a Kirchhoff-type problem involving several potentials and critical nonlinearities on unbounded domain. The first main result is that the problem possesses a ground state solution concentrating around a concrete set relates to the potentials. The second one is that with the help of Nehari manifold and Ljusternik–Schnirelmann category, the relationship between the number of positive solutions and the category of the global minima set of a suitable ground energy function is established. Moreover, we present a sufficient condition for the nonexistence of ground state solution.

Ground states for a coupled Schr\xf6dinger system with general nonlinearities

We study a coupled Schrödinger system with general nonlinearities. By using variational methods, we prove the existence and asymptotic behaviour of ground state solution for the system with periodic couplings. Moreover, we prove the existence and nonexistence of ground state solution for the system with non-periodic couplings via Nehari manifold method. Especially, the ground state solution with both nontrivial components is obtained, and the sign of nontrivial components is considered.

Ground state solutions for the nonlinear Schrödinger‐Poisson systems with sum of periodic and vanishing potentials

We study the existence of ground state solutions for the following Schrödinger‐Poisson equations: urn:x-wiley:mma:media:mma4602:mma4602-math-0001 where is the sum of a periodic potential Vp and a localized potential Vloc and f satisfies the subcritical or critical growth. Although the Nehari‐type monotonicity assumption on f is not satisfied in the subcritical case, we obtain the existence of a ground state solution as a minimizer of the energy functional on Nehari manifold. Moreover, we show that the existence and nonexistence of ground state solutions are dependent on the sign of Vloc.

Existence and Nonexistence of Ground State Solutions to Singular Elliptic Systems

In this paper, a system of semi-linear elliptic equations is investigated, which involves multiple critical nonlinearities and multiple singular points. By variational methods, the existence and nonexistence of minimizers to the Rayleigh quotient and ground state solutions to the system are established.

Ground state solutions to biharmonic equations involving critical nonlinearities and multiple singular potentials

In this paper, a biharmonic problem is investigated, which involves critical Sobolev nonlinearity and multiple Rellich-type terms. By complicated asymptotic analysis and variational arguments, the existence and nonexistence of ground state solution to the problem are established.

Existence and multiplicity solutions of fractional Schrödinger equation with competing potential functions

In this paper, we study the existence and multiplicity of solutions for the following fractional Schrödinger equationwhere h is a positive parameter, , ; denotes the fractional Laplacian of order s, , V, K and Q are competing functions. We first prove that the equation has a nontrivial nonnegative ground state solution and then investigate the relation between the number of solutions and the topology of the global minima set of a suitable ground energy function for sufficiently small. Then, we obtain a sufficient condition for the nonexistence of ground state solution. Finally, we establish an embedding theorem involving fractional weighted Sobolev spaces and use it to get a ground state solution under the case V, K and Q may decay to zero as .

Positive solutions for a class of quasilinear problems with critical growth in ℝN

In this paper, we study the existence, multiplicity and concentration of positive solutions for a class of quasilinear problemswhere —Δp is the p-Laplacian operator for is a small parameter, f(u) is a superlinear and subcritical nonlinearity that is continuous in u. Using a variational method, we first prove that for sufficiently small ε &gt; 0 the system has a positive ground state solution uε with some concentration phenomena as ε → 0. Then, by the minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials. Finally, we obtain some sufficient conditions for the non-existence of ground state solutions.

Semilinear elliptic systems involving multiple critical exponents and singularities in $\mathbb{R}^N$

In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and singular points. By variational methods and analytic techniques,the best constant corresponding to the system is investigated, andthe existence and nonexistence of ground state solutions to thesystem are established.

Existence and nonexistence of the ground state solutions for nonlinear Schrödinger equations with nonperiodic nonlinearities

AbstractIn this paper we study the following nonlinear Schrödinger equation where a is periodic with respect to x and f(x, u) is superlinear in u. Suppose that 0 lies in a gap of the spectrum σ( − Δ + a). Without periodic assumption on f, we prove the existence of ground state solution for the system. Moreover, we obtain some sufficient conditions for the nonexistence of ground state solution.

Existence and nonexistence of ground state solutions for quasilinear elliptic equations with a convection term

By means of lower and supper solutions method, we prove the existence and non-existence of ground state solutions for quasilinear elliptic equations with a convection term. Extending previous results of Goncalves and Silva (2010).

Existence and nonexistence of ground state solutions for elliptic equations with a convection term

We deal with the existence of positive solutions u decaying to zero at infinity, for a class of equations of Lane–Emden–Fowler type involving a gradient term. One of the main points is that the differential equation contains a semilinear term σ ( u ) where σ : ( 0 , ∞ ) → ( 0 , ∞ ) is a smooth function which can be both unbounded at infinity and singular at zero. Our technique explores symmetry arguments as well as lower and upper solutions.

Ground State of N Coupled Nonlinear Schrodinger Equations in $${\mathbb{R}}^n, n \leq 3$$

We establish some general theorems for the existence and nonexistence of ground state solutions of steady-state N coupled nonlinear Schrodinger equations. The sign of coupling constants β ij ’s is crucial for the existence of ground state solutions. When all β ij ’s are positive and the matrix Σ is positively definite, there exists a ground state solution which is radially symmetric. However, if all β ij ’s are negative, or one of β ij ’s is negative and the matrix Σ is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N=3.

Ground State of N Coupled Nonlinear Schr�dinger Equations in Rn,n?3

We establish some general theorems for the existence and nonexistence of ground state solutions of steady-state N coupled nonlinear Schrodinger equations. The sign of coupling constants βij’s is crucial for the existence of ground state solutions. When all βij’s are positive and the matrix Σ is positively definite, there exists a ground state solution which is radially symmetric. However, if all βij’s are negative, or one of βij’s is negative and the matrix Σ is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N=3.