We study the following zero-mass Schr & ouml;dinger-Poisson-Slater equation ( 1 ) -Delta u + 4 pi|x| * u2 u = f (|x|, u), u is an element of D1,2(R3) with nonlinearity subscaled near zero in the sense that f (|x|, t) approximate to a|t|p-2t as |t| -> 0 for some p is an element of ( 7 18 , 3). A nonzero solution is obtained via Morse theory when the nonlinearity is asymptotically scaled at infinity. For this purpose we prove an abstract result on the critical groups at infinity for functionals satisfying the geometric assumptions of the scaled saddle point theorem of Mercuri and Perera (2024) [20]. For the case that f (|x|, .) is odd, a sequence of solutions is obtained via a version of Clark's theorem due to Kajikiya (2005) [11]. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper we consider nonlinear biharmonic equations with p-Laplacian (p≥2) of the form {[ Δ^2 u - Δ_p u + V (x) u = f (x, u) , u ∈ H^2 (ℝ^N) , ]. where the potential V(x) may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity f(x,·) is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that f(x,u)=λg (x) | u |^q - 2 u + | u |^m - 2 u with exponent m critical or subcritical.

We obtain multiple solutions for the zero mass Schrödinger–Poisson–Slater equation - Δ u + ( 1/4 π | x | *u^2 ) u = λ g (x) | u |^p - 2 u + | u |^6 - 2 u , u ∈𝒟^1, 2 (ℝ^3) for λ≫ 1 , where p ∈ (4, 6) and g ∈ L^6 / (6 - p) (ℝ^3) . The crucial (PS)_c condition is verified using a simpler method. Similar multiplicity result is also obtained for related equation with an external potential.

It is known that the integral of the Jacobian determinant of a smooth map f: →ℝ^n depends only on f |_∂ Ω and this result leads to an analytic proof of the Brouwer fixed point theorem. In this note we provide two new proofs of this result, one by classical analysis and one by differential forms and Stokes formula.

Let p, q be functions on ℝ^N satisfying 1≪ q≪ p≪ N , we consider p(x)-Laplacian problems of the form {[ -Δ _p(x)u+V(x)| u| ^p(x)-2u=λ| u| ^q(x)-2u+g(x,u),; u∈ W^1,p(x)(ℝ^N). ]. To apply variational methods, we introduce a subspace X of W^1,p(x)(ℝ^N) as our working space. Compact embedding from X into L^q(x)(ℝ^N) is established, this enable us to get nontrivial solution of the problem; and two sequences of solutions going to ∞ and 0 respectively, when g(x,· ) is odd.

Some existence results of periodic solution are obtained for a class of second-order Hamiltonian systems with nonlinearity depending on derivative. We prove that there exists T-0 > 0 such that, for any T < T-0 , the provided Hamiltonian system has a nontrivial T-periodic and T /2-antiperiodic solution via linking theorem and iteration method.

In this paper we study the following ( p,q) -Laplacian equation with critical exponent -Δ _pu-Δ _qu=λ h(x)|u|^r-2u+g(x)|u|^p^* -2u in ℝ^N , where 1

By applying Clark’s theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(u)inR3,−Δϕ=u2inR3 with negative energy. A similar result is also obtained for the Schrödinger-Kirchhoff equation as follows:−1+∫RN∇u2Δu+V(x)u=f(u)u∈H1(RN).

This paper is concerned with the quasilinear Schrodinger equation -delta u + V(x)u - delta[(1 + u(2))(1/2)]u/2(1 + u(2))(1/2) = h(x, u), in R-N, where N >= 3, V is a given potential allowed to be indefinite, or equivalently, the Schrodinger operator -delta + V can be indefinite. We consider the case that V is coercive so that the working space can be compactly embedded into Lebesgue spaces. Using the local linking theorem and Morse theory, we obtain a nontrivial solution for the above problem. Moreover, by the symmetric mountain pass theorem, we get an unbounded sequence of solutions.

We consider the Dirichlet problem for p(x)-Laplacian equations of the form -Delta(p(x))u+b(x)|u|(p(x)-2)u=f(x,u),u is an element of W(1,p(x))0(Omega). The odd nonlinearityf(x,u)isp(x)-sublinear atu=0 but the related limit need not be uniform for x is an element of Omega.Except being subcritical, no additional assumption is imposed on f(x,u)for|u| large. By applying Clark'stheorem and a truncation method, we obtain a sequence of solutions with negative energy and approachingthe zero functionu=0

In this paper we consider 6-superlinear Chern–Simons–Schrödinger systems. In contrast to most studies, we consider the case where the potential V is indefinite so that the Schrödinger operator −Δ+V possesses a finite-dimensional negative space. We obtain nontrivial solutions for the problem via Morse theory.

Abstract This paper presents an induction proof of the G-N-S inequality when p = 1, which the author believes is much simpler and more transparent than other proofs found in the literature.

A bstract . We obtain a sequence of solutions converging to zero for the Kirchho ff equation

We obtain a sequence of solutions converging to zero for the Kirchhoff equation -( 1+∫_Ω|∇ u|^2) Δ u+V(x)u=f(u), u∈ H_0^1(Ω) via truncating technique and a variant of Clark's theorem due to Liu–Wang, where Ω is a bounded smooth domain Ω⊂ℝ^N. Similar result for Schrödinger-Poisson system on a bounded smooth domain Ω⊂ℝ^3 is also presented.

In this paper, we consider the following quasilinear Schrödinger equation $$\begin{aligned} -\Delta u-u\Delta (u^{2})=k(x)\left| u\right| ^{q-2}u-h(x)\left| u\right| ^{s-2}u,u\in D^{1,2}(\mathbb {R}^{N}), \end{aligned}$$ where $$1"2\cdot 2^{*}$$ . By taking advantage of geometric properties of a nonlinear transformation f and a variant of Clark’s theorem, we get a sequence of solutions with negative energy in a space smaller than $$D^{1,2}(\mathbb {R}^{N})$$ . Nonnegative solution at negative energy level is also obtained."

We study a class of Schrödinger–Kirchhoff equations in R3 with subquadratic nonlinearity and indefinite potential. By variational methods, we obtain the existence of multiple solutions for this problem.

We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(\phi(|\nabla u|)\nabla u\big)+V(x)\phi (|u|)u=f(u)\qquad u\in W^{1,\Phi}(\mathbb{R}^{N}), \] where $\phi$ and $f$ satisfy suitable conditions. The positive potential $V\in C(\mathbb{R}^{N})$ exhibits a finite or infinite potential well in the sense that $V(x)$ tends to its supremum $V_{\infty}\le+\infty$ as $|x|\to\infty$. Nontrivial solutions are obtained by variational methods. When $V_{\infty }=+\infty$, a compact embedding from a suitable subspace of $W^{1,\Phi }(\mathbb{R}^{N})$ into $L^{\Phi}(\mathbb{R}^{N})$ is established, which enables us to get infinitely many solutions for the case that $f$ is odd. For the case that $V(x)=\lambda a(x) + 1$ exhibits a steep potential well controlled by a positive parameter $\lambda$, we get nontrivial solutions for large $\lambda$.

介绍了在我校数学系二年级第一学期的本科生讲授多元微积分的一些做法.特别强调向量值函数的微分学和将实际问题转化为积分的微元分析法,且举例说明如何把学生已掌握的线性代数和常微分方程知识引入多元微积分中来,得到有重要意义的结果.

We consider a class of stationary Schrödinger-Poisson systems with a general nonlinearity f(u) and coercive sign-changing potential V so that the Schrödinger operator −Δ+V is indefinite. Previous results in this framework required f to be strictly 3-superlinear, thus missing the paramount case of the Gross-Pitaevskii-Poisson system, where f(t)=|t|2t; in this paper we fill this gap, obtaining non-trivial solutions when f is not necessarily 3-superlinear.

We obtain existence and multiplicity results for fourth order elliptic equations on RN involving uΔ(u2) and sign-changing potentials. Our results generalize some recent results on this kind of problems. To study this kind of problems, we first consider the case that the potential V is coercive so that the working space can be compactly embedded into Lebesgue spaces. Then we studied the case that the potential V is bounded so that the working space is exactly H2(RN), which can not be compactly embedded into Lebesgue spaces anymore. To deal with this more difficult case, we study the weak continuity of the term in the energy functional corresponding to the term uΔ(u2) in the equation.