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We deal with localized semiclassical states for singularly perturbed Kirchhoff-type equation as follows: −ε2a+εb∫R3∣∇v∣2dxΔv+V(x)v=P(x)f(v),x∈R3,-\left({\varepsilon }^{2}a+\varepsilon b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}| \nabla v{| }^{2}{\rm{d}}x\right)\Delta v+V\left(x)v=P\left(x)f\left(v),\hspace{1em}x\in {{\mathbb{R}}}^{3}, where V,P∈C1(R3,R)V,P\in {C}^{1}\left({{\mathbb{R}}}^{3},{\mathbb{R}}) and bounded away from zero. By applying the penalization approach together with the Nehari manifold approach in the studies of Szulkin and Weth, we obtain the existence of an infinite sequence of localized solutions of higher topological type. In addition, we also give a concrete set as the concentration position of these localized solutions. It is noted that, in our main results, ff only belongs to C(R,R)C\left({\mathbb{R}},{\mathbb{R}}) and does not satisfy the Ambrosetti-Rabinowitz-type condition.

In this paper, we consider the following critical equation: − Δ u + V (y) u = K (y) u N + 2 N − 2 , u > 0 , u ∈ H 1 (ℝ N) , where (y ′ , y ″) ∈ ℝ 2 × ℝ N − 2 , V (| y ′ | , y ″) and K (| y ′ | , y ″) are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if N ≥ 5 , K (r , y ″) has a stable critical point (r 0 , y 0 ″) with r 0 > 0 , K (r 0 , y 0 ″) > 0 and B 1 : = V (r 0 , y 0 ″) ∫ ℝ N U 0 , 1 2 d y − Δ K (r 0 , y 0 ″) 2 ∗ N ∫ ℝ N | y | 2 U 0 , 1 2 ∗ d y > 0 , then the above equation has infinitely many positive solutions, where U 0 , 1 is the unique positive solution of − Δ u = u N + 2 N − 2 with u (0) = max y ∈ ℝ N u (y). Combining the results of [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations; S. Peng, C. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal.274 (2018) 2606–2633], it implies that the role of stable critical points of K (r , y ″) in constructing bump solutions is more important than that of V (r , y ″) and that V (r 0 , y 0 ″) can influence the sign of Δ K (r 0 , y 0 ″) , i.e. Δ K (r 0 , y 0 ″) can be nonnegative, different from that in [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations]. The concentration points of the solutions locate near the stable critical points of K (r , y ″) , which include the case of a saddle point. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear fractional Schrödinger-Poisson system where s, t ∈ (0, 1) and 4s + 2t ≥ 3, 0 < q < 1, and a, K, V ∈ L∞(ℝ3). When a, V both change sign in ℝ3, by applying the symmetric mountain pass theorem, we prove that the problem has infinitely many solutions under appropriate assumptions on a, K, V. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of nontrivial solution for a class of quasilinear Schrödinger equations in ${\mathbb{R}}$ with the nonlinearity asymptotically linear and, furthermore, the potential indefinite in sign. The tool used in this paper is the direct variation method. [ABSTRACT FROM AUTHOR]