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In this article, we consider the following problem: - Δ 픹 N ⁢ u - λ ⁢ u = | u | p - 1 ⁢ u , u ∈ H 1 ⁢ ( 픹 N ) , -\Delta_{\mathbb{B}^{N}}u-\lambda u=|u|^{p-1}u,\quad u\in H^{1}(\mathbb{B}^{N}), where 픹 N {\mathbb{B}^{N}} represents the Poincaré ball model of the hyperbolic space, 1 < p < 2 * - 1 = N + 2 N - 2 {1

In the present paper, we investigate the following nonlinear Kirchhoff-type equation with Sobolev exponent$ \begin{align} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\, \, dx\right)\Delta u+\lambda V(x)u = |u|^{4}u + |u|^{p-1}u, \qquad x\in \mathbb{R}^{3}, \end{align} $where $ a, b>0 $, $ 4

In this article we establish the existence of a positive solution and multiple sign-changing solutions to the following supercritical exponent problem involving fractional Laplace operator (P b , Ω s) (- Δ) s u = b (x) | u | q - 2 u in Ω , u = 0 on ∂ Ω , where Ω is a bounded smooth domain with non-trivial topology in R N which is invariant under a group G of orthogonal transformations of R N , N ≥ 2 s , s ∈ (0 , 1) , b ∈ C 0 , α (Ω ¯) and positive, q > 2 s ∗ : = 2 N N - 2 s . [ABSTRACT FROM AUTHOR]

We prove the existence of sign-changing solution to the problem - Δ u - 1 2 x · ∇ u = λ u , in R + N , ∂ u ∂ ν = | u | 2 ∗ - 2 u , on ∂ R + N , where R + N = { (x ′ , x N) : x ′ ∈ R N - 1 , x N > 0 } is the upper half-space, 2 ∗ : = 2 (N - 1) / (N - 2) , N ≥ 7 , ∂ u ∂ ν is the partial outward normal derivative and the parameter λ > 0 interacts with the spectrum of the linearized problem. In the proof, we apply variational methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff-type problem with critical nonlinearity - a + b ∫ Ω | ∇ u | 2 d x Δ u = λ f (x) | u | p - 2 u + | u | 4 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R 3 , a > 0 is a constant, b , λ are positive parameters and 2 < p < 4 . Under different assumptions on the nonlinearity, the equation has been extensively considered in the case 4 < p < 6 . By contrast, there is no existence result of solutions for the case 2 < p < 4 since the appearance of the nonlocal term. By using some innovative analytical skills, we obtain the existence results about the sign-changing solutions of this problem. Furthermore, we also present asymptotic behaviors of the sign-changing solutions as b ↘ 0 or λ ↘ 0 . [ABSTRACT FROM AUTHOR]

We show the existence of unbounded connected components of 2π-periodic positive solutions for the equations with one-dimensional Minkowski-curvature operator − u ′ 1 − u ′ 2 ′ = λ a (x) f (u , u ′ ) , x ∈ R , where λ > 0 is a parameter, a ∈ C (R , R) is a 2π-periodic sign-changing function with ∫ 0 2 π a (x) d x < 0 , f ∈ C (R × R , R) satisfies a generalized regular-oscillation condition. Moreover, for the special case that f does not contain derivative term, we also investigate the global structure of 2π-periodic odd/even sign-changing solutions set under some parity conditions. The proof of our main results are based upon bifurcation techniques. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Schrödinger-Poisson system with a critical nonlinearity and a critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality$ \begin{equation*} \begin{cases} -\Delta u +u +\lambda\phi|u|^3u = |u|^4u +|u|^{q-2}u, \ \ &\ x\in\mathbb{R}^3,\\ -\Delta\phi = |u|^5, \ \ &\ x\in\mathbb{R}^3, \end{cases} \end{equation*} $where $ \lambda\in \mathbb{R} $ is a parameter and $ q\in(2,6) $. By employing variational methods, we present the existence and nonexistence results (depending on the parameters $ \lambda $ and $ q $) as follows:($ i $) if $ \lambda\ge (\frac{q+2}{8})^2 $ and $ q\in(2,6) $, there is no nontrivial solution of the above system;($ ii $) if $ \lambda\in (-\lambda^*,0] $ for some $ \lambda^*>0 $, we obtain a least energy radial sign-changing solution $ u_\lambda $ of the above system;($ iii $) we consider $ \lambda $ as a parameter and analyze the asymptotic behavior of $ u_\lambda $ as $ \lambda\to 0^- $.Moreover, a Schrödinger-Poisson system with a critical nonlocal term differs completely from a system with a subcritical nonlocal term. [ABSTRACT FROM AUTHOR]

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L ∞ and in the Musielak–Orlicz Sobolev space. [ABSTRACT FROM AUTHOR]

This paper deals with the following fractional Laplacian system with critical exponent: (−Δ)su=λ1u+μ1|u|2s∗−2u+β|u|2s∗2−2u|v|2s∗2,inΩ,(−Δ)sv=λ2v+μ2|v|2s∗−2v+β|v|2s∗2−2v|u|2s∗2,inΩ,u=v=0,inRN\Ω,$$ \left\{\begin{array}{ll}{\left(-\Delta \right)}&#x0005E;su&#x0003D;{\lambda}_1u&#x0002B;{\mu}_1{\left&#x0007C;u\right&#x0007C;}&#x0005E;{2_s&#x0005E;{\ast }-2}u&#x0002B;\beta {\left&#x0007C;u\right&#x0007C;}&#x0005E;{\frac{2_s&#x0005E;{\ast }}{2}-2}u{\left&#x0007C;v\right&#x0007C;}&#x0005E;{\frac{2_s&#x0005E;{\ast }}{2}},\kern0.30em & \mathrm{in}\kern3.0235pt \hspace{0.1em}\Omega, \\ {}{\left(-\Delta \right)}&#x0005E;sv&#x0003D;{\lambda}_2v&#x0002B;{\mu}_2{\left&#x0007C;v\right&#x0007C;}&#x0005E;{2_s&#x0005E;{\ast }-2}v&#x0002B;\beta {\left&#x0007C;v\right&#x0007C;}&#x0005E;{\frac{2_s&#x0005E;{\ast }}{2}-2}v{\left&#x0007C;u\right&#x0007C;}&#x0005E;{\frac{2_s&#x0005E;{\ast }}{2}},\kern0.30em & \mathrm{in}\kern3.0235pt \hspace{0.1em}\Omega, \\ {}u&#x0003D;v=0,& \mathrm{in}\kern3.0235pt \hspace{0.1em}{\mathfrak{R}}&#x0005E;N\backslash \Omega, \end{array}\right. $$where Ω$$ \Omega $$ is a bounded smooth open connected set in RN,2s∗=2NN−2s$$ {\mathfrak{R}}&#x0005E;N,{2}_s&#x0005E;{\ast }&#x0003D;\frac{2N}{N-2s} $$ is the fractional critical Sobolev exponent, s∈(0,1),μ1,μ2>0,0<λ1,λ2<λ1(Ω)$$ s\in \left(0,1\right),{\mu}_1,{\mu}_2>0,0<{\lambda}_1,{\lambda}_2<{\lambda}_1\left(\Omega \right) $$ and λ1(Ω)$$ {\lambda}_1\left(\Omega \right) $$ is the first eigenvalue of fractional Laplacian (−Δ)s$$ {\left(-\Delta \right)}&#x0005E;s $$ under the condition u=0$$ u&#x0003D;0 $$ in RN\Ω$$ {\mathfrak{R}}&#x0005E;N\backslash \Omega $$. We prove that, for each fixed β∈−μ1μ2,0$$ \beta \in \left(-\sqrt{\mu_1{\mu}_2},0\right) $$ and λ1,λ2$$ {\lambda}_1,{\lambda}_2 $$ slightly smaller than λ1(Ω)$$ {\lambda}_1\left(\Omega \right) $$, the above system with N≥5s$$ N\ge 5s $$ admits a sign‐changing solution in the following sense: one component changes sign, while the other one is positive. Our result includes the lower dimensional case N∈[5s,6s)$$ N\in \left[5s,6s\right) $$. Compared with the classical Laplacian case, our problem is nonlocal and the first component of the solution is sign‐changing, some new difficulties arise and new arguments and estimates should be introduced. [ABSTRACT FROM AUTHOR]

We study the optimal partition problem for the prescribed constant Q-curvature equation induced by the higher-order conformal operators under the effect of cohomogeneity one actions on Einstein manifolds with positive scalar curvature. This allows us to give a precise description of the solution domains and their boundaries in terms of the orbits of the action. We also prove the existence of least energy symmetric solutions to a weakly coupled elliptic system of prescribed Q-curvature equations under weaker assumptions and conclude a multiplicity result of sign-changing solutions to the prescribed constant Q-curvature problem induced by the Paneitz-Branson operator. Moreover, we study the coercivity of GJMS operators on Ricci solitons, compute the Q-curvature of these manifolds, and give a multiplicity result for the sign-changing solutions to the Yamabe problem with a prescribed number of nodal domains on the Koiso–Cao Ricci soliton. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following N-Laplacian Kirchhoff-type problem with logarithmic and exponential nonlinearities: - a + b ∫ Ω | ∇ u | N d x Δ N u = | u | p - 2 u ln | u | 2 + λ f (u) , in Ω , u = 0 , on ∂ Ω , where f(t) behaves like e x p α | t | N N - 1 . Combining constrained variational method, topological degree theory and quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution u b with precisely two nodal domains. Moreover, we show that the energy of u b is strictly larger than two times of the ground state energy and analyze the convergence property of u b as b ↘ 0 . [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional $ p\&q $ p &q -Laplacian equation: \[ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right)=K(x) f(u) \quad {\rm in}\ \mathbb{R}^{N}, \] (− Δ) p s u + (− Δ) q s u + V (x) (| u | p − 2 u + | u | q − 2 u) = K (x) f (u) in R N , where $ s \in (0,1), 1 \lt p \lt q \lt \frac {N}{s} $ s ∈ (0 , 1) , 1 < p < q < N s , and $ (-\Delta)_{t}^{s} $ (− Δ) t s with $ t \in \{p, q\} $ t ∈ { p , q } is the fractional t-Laplacian operator, f is a $ \mathcal {C}^1 $ C 1 real function and V, K are continuous, positive functions. By using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove the existence of a least energy sign-changing solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger-Poisson system $ \begin{equation*} \qquad \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u = |u|^{p-2}u+ \lambda K(x)|u|^{q-2}u\ \ \ &\ \rm in\; \mathbb{R}^{3}, \\ -\Delta \phi = u^2 \ \ \ &\ \rm in\; \mathbb{R}^{3}.\ \end{array} \right. \end{equation*} $ Under the weakly coercive assumption on $ V $ and an appropriate condition on $ K $, we investigate the cases when the nonlinearities are of concave-convex type, that is, $ 1 < q < 2 $ and $ 4 < p < 6 $. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that $ \lambda\in(-\infty, \lambda_*) $, where $ \lambda_* > 0 $ is a constant.

In this study, we investigate the existence of infinitely many radial sign-changing solutions with nodal properties for a class of Kirchhoff-type equations possessing non-odd nonlinearity. By combining variational methods and analysis techniques, we prove that for any positive integer k, the equation has a radial solution that changes signs exactly k times. Furthermore, we demonstrate that the energy of such solutions is an increasing function of k. Owing to the inherent characteristics of these equations, the methods used herein significantly differ from those used in the existing literature. Particularly, we discover a unified method to obtain infinitely many radial sign-changing solutions with nodal properties for local and nonlocal problems. [ABSTRACT FROM AUTHOR]

In the present paper, we study the Hénon type parabolic equation $ \phi_t-\Delta \phi = |x|^\beta |\phi|^{\alpha-1}\phi, x\in\mathbb{R}^{N} $ with singular initial data $ \eta|x|^{-\frac{2+\beta}{\alpha-1}} $, where $ \eta, \beta>0, \alpha>1 $. We prove the existence of infinitely many sign-changing self-similar solutions by analyzing the related inverted profile equation for $ N\geq1 $. This can be seen as an extension of the recent work of T. Cazenave at al.(Amer. J. Math. 142 (2020), 1439-1495) to the Hénon type parabolic equation. [ABSTRACT FROM AUTHOR]

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We study the existence of sign-changing solutions for the following supercritical problem with variable exponent and logarithmic nonlinearity$ \begin{equation*} \left\{ \begin{aligned} - \Delta u& = |u|^{2^{*}-2} u\big(\ln(\tau +|u|) \big)^{ |x|^{\beta} } & \ \ \mbox{in} \ \ \ B_{1}, \\ u& = 0 \ & \mbox{on} \ \partial B_{1}, \end{aligned} \right. \end{equation*} $where $ B_{1} $ is the unit ball in $ \mathbb{R}^{N} $, $ N\geq 3 $, $ 2^* = 2N/(N-2) $ is the critical Sobolev exponent, and $ \tau\geq 1 $, $ \beta>0 $ are constants. For any $ k\in \mathbb{N} $, if $ \tau\geq 1 $ and $ 0 <\beta<(N-2)/2 $, we show that there exist one pair of solutions which change sign exactly $ k $ times by variational methods. [ABSTRACT FROM AUTHOR]

In this paper, based on a suitable fractional Trudinger–Moser inequality, we establish sufficient conditions for the existence result of least energy sign-changing solution for a class of one-dimensional nonlocal equations involving logarithmic and exponential nonlinearities. By using a main tool of constrained minimization in Nehari manifold and a quantitative deformation lemma, we consider both subcritical and critical exponential growths. This work can be regarded as the complement for some results of the literature.

We are concerned with sign-changing solutions of the following gauged nonlinear Schrödinger systems in dimension two including the so-called Chern-Simons terms$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u_{1}+\omega u_{1}+\big(\frac{h_{1}^{2}(|x|)}{|x|^{2}}+\int_{|x|}^{+ \infty}\frac{h_{1}(s)}{s}u_{1}^{2}(s) \mathrm{d} s\big)u_{1}\\=|u_{1}|^{2p-2}u_{1}+b|u_{2}|^{p}|u_{1}|^{p-2}u_{1}&\mbox{in}\ \mathbb R^2,\\ -\Delta u_{2}+\omega u_{2}+\big(\frac{h_{2}^{2}(|x|)}{|x|^{2}}+\int_{|x|}^{+ \infty}\frac{h_{2}(s)}{s}u_{2}^{2}(s) \mathrm{d} s\big)u_{2}\\=|u_{2}|^{2p-2}u_{2}+b|u_{1}|^{p}|u_{2}|^{p-2}u_{2}&\mbox{in}\ \mathbb R^2,\\ u_{i}(x)=u_{i}(|x|)\in H^{1}(\mathbb R^{2}),i=1,2,\\ \end{array} \right. \end{eqnarray*} $where $ \omega>0, p>3, b<0 $ and$ h_{1}(s)=\frac{1}{2}\int_0^{s}\tau u_{1}^{2}(\tau) \mathrm{d} \tau, \, \, h_{2}(s)=\frac{1}{2}\int_0^{s}\tau u_{2}^{2}(\tau) \mathrm{d} \tau.\ $Combining the method of invariant sets of descending flow and a general critical point theory, we investigate the existence and multiplicity of sign-changing solutions to the system above. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger-Poisson system { − Δ u + V (x) u + ϕ u = | u | p − 2 u + λ K (x) | u | q − 2 u i n R 3 , − Δ ϕ = u 2 i n R 3. Under the weakly coercive assumption on V and an appropriate condition on K , we investigate the cases when the nonlinearities are of concave-convex type, that is, 1 < q < 2 and 4 < p < 6 . By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that λ ∈ (− ∞ , λ ∗) , where λ ∗ > 0 is a constant. In this paper, we consider the following Schrödinger-Poisson system Under the weakly coercive assumption on and an appropriate condition on , we investigate the cases when the nonlinearities are of concave-convex type, that is, and . By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that , where is a constant. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following quasilinear Kirchhoff–Schrödinger–Poisson system: { − (a + b ∫ R 3 | ∇ u | 2 d x) Δ u + V (x) u − 1 2 u Δ (u 2) + ϕ u = g (u) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , where a, b are positive constants, V (x) is a smooth potential function and g is an appropriate nonlinear function. To overcome the technical difficulties caused by the quasilinear term, the perturbation method of adding 4-Laplacian operator is adapted to consider the perturbation problem, so that the corresponding functional has both smoothness and compactness in the appropriate space. Moreover, when g satisfies the appropriate hypotheses, a sign-changing solution u 0 of above problem can be obtained by taking advantage of constraint variational method, the quantitative deformation lemma and approximation technique, which has precisely two nodal domains. [ABSTRACT FROM AUTHOR]

In this paper, the following new class of problems is studied where Ω is a smooth bounded domain in R N ( N ≥ 3 ), p ∈ C (Ω ¯) such that 2 ≤ p (x) < N for all x ∈ Ω ¯ , and a is C 1 real function. Without assuming the C 1 and the well-known Ambrosetti-Rabinowitz conditions on the reaction term f, we prove that problem P has a least-energy sign-changing weak solution. Note that, our approach is based on a variant of the quantitative deformation lemma and some new technical lemmas. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schr¨odinger-Poisson system  -Δu + V(x)u + ϕu = |u|p-2u + λK(x)|u|q-2u in R3, -Δϕ = u2 in R3. Under the weakly coercive assumption on V and an appropriate condition on K, we investigate the cases when the nonlinearities are of concave-convex type, that is, 1 < q < 2 and 4 < p < 6. By constructing a nonempty closed subset of the sign-changing Nehari manifold, we establish the existence of least energy sign-changing solutions provided that λ ∈ (-∞, λ∗), where λ∗ > 0 is a constant. [ABSTRACT FROM AUTHOR]

This paper considers the following fractional $ (p, q) $-Laplacian equation: $ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right) = \lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $ where $ s \in(0, 1), \lambda > 0, 2 < p < q < \frac{N}{s} $, $ (-\Delta)_{t}^{s} $ with $ t \in\{p, q\} $ is the fractional $ t $-Laplacian operator, and potential $ V $ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $ u_{\lambda} $ under suitable conditions on $ f $, $ V $ and $ \lambda $. Moreover, we show that the energy of $ u_{\lambda} $ is strictly larger than two times the ground state energy.

We investigate the following nonlinear system $ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u+\phi u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ -\Delta\phi = u^{2}, \; \ x\in\Omega, \\ u = \phi = 0, \; \ x\in \partial\Omega, \end{cases} $ with $ a, b > 0 $, $ \lambda, \mu\in\mathbb{R} $, and $ \Omega\subset \mathbb{R}^{3} $ is bounded with smooth boundary. Let $ \lambda_{1} > 0 $ be the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. We get that for certain $ \widetilde{\mu} > 0 $ there exists at least one least energy sign-changing solution for the above system if $ \lambda < a\lambda_{1} $ and $ \mu > \widetilde{\mu} $. In addition, we remark that the nonlinearity $ \lambda u+\mu|u|^{2}u $ does not satisfy the growth conditions.

This paper focuses on a class of supercritical, quasi-linear Schrödinger equations. Based on the methods of invariant sets, some results about the existence and multiplicity of sign-changing solutions for supercritical equations are obtained.

We are interested in the existence of sign-changing solutions for the following Kirchhoff-type equation - a + b ∫ Ω | ∇ u | 2 d x Δ u = h + (x) + λ h - (x) | u | p - 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where a , b > 0 , Ω ⊂ R 3 is a bounded domain with smooth boundary, the potential h : Ω ¯ → R is a sign-changing continuous function, and λ > 0 is a parameter. If p ∈ (4 , 6) , we prove the existence of least energy sign-changing solution u b , λ , the asymptotic behavior of u b , λ as b → 0 + or λ → + ∞ are also analyzed. Moreover, if the set { x ∈ Ω : h (x) > 0 } possesses several disjoint components, we also prove the existence of multi-bump sign-changing solutions. [ABSTRACT FROM AUTHOR]

We consider a parametric Neumann problem driven by a nonhomogeneous differential operator, with a reaction consisting of a critical term plus a Carathéodory perturbation which is only locally restricted near zero and is not assumed to be odd. Combining variational methods, with upper-lower solutions and truncation techniques, and flow invariance arguments, we show that the problem admits infinitely many sign-changing solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of a least energy sign-changing solution for the following Schrödinger-Kirchhoff equation SK - (a + b ∫ R 3 | ∇ u | 2 d x) Δ u + V (x) u = f (x , u) , x ∈ R 3 , u ∈ H 1 (R 3) where a , b > 0 are parameters, V ∈ C (R 3 , R +) and f ∈ C (R 3 × R , R) . The potential function V satisfies some suitable conditions and the nonlinearity f satisfies mild assumptions. By using sign-changing Nehari manifold, we prove that this problem possesses a ground state sign-changing solution with precisely two nodal domains, and its energy is strictly larger than twice that of ground state solutions. Besides, we obtain a convergence property of u b n as b n ↘ 0 . Our results unify asymptotically cubic, which enrich and improve the previous ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the sign-changing solutions to the following Schrödinger-Poisson system - Δ u + V (x) u + λ ϕ (x) u = f (u) , x ∈ R 3 , - Δ ϕ = u 2 , x ∈ R 3 , where λ > 0 is a parameter and f is super 2-linear at infinity. By using the method of invariant sets of descending flow and a multiple critical points theorem, we prove that this system possesses infinitely many sign-changing solutions for any λ > 0 . [ABSTRACT FROM AUTHOR]

This paper considers the following fractional -Laplacian equation: where , with is the fractional -Laplacian operator, and potential is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution under suitable conditions on , and . Moreover, we show that the energy of is strictly larger than two times the ground state energy. [ABSTRACT FROM AUTHOR]

We investigate the following nonlinear system { − (a + b ∫ Ω | ∇ u | 2 d x) Δ u + ϕ u = λ u + μ | u | 2 u , x ∈ Ω , − Δ ϕ = u 2 , x ∈ Ω , u = ϕ = 0 , x ∈ ∂ Ω , with a , b > 0 , λ , μ ∈ R , and Ω ⊂ R 3 is bounded with smooth boundary. Let λ 1 > 0 be the first eigenvalue of (− Δ u , H 0 1 (Ω)). We get that for certain μ ~ > 0 there exists at least one least energy sign-changing solution for the above system if λ < a λ 1 and μ > μ ~ . In addition, we remark that the nonlinearity λ u + μ | u | 2 u does not satisfy the growth conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation − (a + b ∫ V | ∇ u | 2 d μ) Δ u + c (x) u = f (u) on a locally finite graph G = (V , E) , where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution u b of the above equation if c (x) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c (x) u = f (u). [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence and multiplicity of r a d i a l sign-changing solutions for fractional Schrödinger–Poisson system: (℘) (− Δ) s u + u + ϕ u = f (u) , (− Δ) α ϕ = u 2 , in R 3 where s ∈ (3 4 , 1) , α ∈ (0 , 1) and f is a continuous function. Based on perturbation approach and the method of invariant sets of descending flow, we obtain the existence and multiplicity of r a d i a l sign-changing solutions of system ( P ). In addition, by applying the constrained variational method incorporated with Brouwer degree theory, we prove that system ( P ) possesses at least one r a d i a l ground state sign-changing solution. Furthermore, we show that the least energy of sign-changing solutions exceed twice than the least energy, and when f is odd, system ( P ) admits infinitely many nontrivial solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider the following Klein–Gordon equation coupled with Born–Infeld theory - Δ u + [ m 2 - (ω + ϕ) 2 ] u = | u | p - 2 u in R 3 , Δ ϕ + β Δ 4 ϕ = 4 π (ω + ϕ) u 2 in R 3 , where 2 < p < 6 , ω > 0 , β > 0 and m is a real constant. Assuming that 0 < ω < p 2 - 1 | m | and 2 < p < 4 or 0 < ω < | m | and 4 ≤ p < 6 , we obtain the existence and multiplicity of sign-changing solutions via the method of invariant sets of descending flow. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence of least energy sign-changing solutions for the following Schrödinger-Poisson system −Δu+V(x)u+K(x)ϕu=f(u),x∈R3,−Δϕ=K(x)u2,x∈R3,\left\{\begin{array}{ll}-\Delta u+V\left(x)u+K\left(x)\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2},\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ \hspace{1.0em}\end{array}\right. where the functions V(x),K(x)V\left(x),K\left(x) have finite limits as ∣x∣→∞| x| \to \infty satisfying some mild assumptions. By combining variational methods with the global compactness lemma, we obtain a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

In this paper we study the asymptotic behavior of solutions to the (p , q) -equation − Δ p u − Δ q u = f (x , u) in Ω , u = 0 on ∂ Ω , as p → 1 + , where N ≥ 2 , 1 < p < q < 1 ∗ ≔ N / (N − 1) and f is a Carathéodory function that grows superlinearly and subcritically. Based on a Nehari manifold treatment, we are able to prove that the (1 , q) -Laplace problem given by − div ∇ u | ∇ u | − Δ q u = f (x , u) in Ω , u = 0 on ∂ Ω , has at least two constant sign solutions and one sign-changing solution, whereby the sign-changing solution has least energy among all sign-changing solutions. Furthermore, the solutions belong to the usual Sobolev space W 0 1 , q (Ω) which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space BV (Ω) of all functions of bounded variation. As far as we know this is the first work dealing with (1 , q) -Laplace problems even in the direction of constant sign solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential (1+b[u]α2)((-Δx)αu-Δyu)+V(x,y)u=f(x,y,u),(x,y)∈ℝN=ℝn×ℝm,\left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alpha }u - {\Delta _y}u} \right) + V\left( {x,y} \right)u = f\left( {x,y,u} \right),\left( {x,y} \right) \in {\mathbb{R}^N} = {\mathbb{R}^n} \times {\mathbb{R}^m}, where [u]α=(∫ℝN(|(-Δx)α2u|2+|∇yu|2)dxdy)12{\left[ u \right]_\alpha } = {\left( {\int {_{{\mathbb{R}^N}}\left( {{{\left| {{{\left( { - {\Delta _x}} \right)}^{{\alpha \over 2}}}u} \right|}^2} + {{\left| {{\nabla _y}u} \right|}^2}} \right)dxdy} } \right)^{{1 \over 2}}}. Based on variational approach and a variant of the quantitative strain lemma, for each b > 0, we show the existence of a least energy nodal solution ub. In addition, a convergence property of ub as b ↘ 0 is established.

We deal with sign-changing solutions for the Kirchhoff equation $ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ u = 0, \; \ x\in \partial\Omega, \end{cases} $ where $ a, b > 0 $ and $ \lambda, \mu\in\mathbb{R} $ being parameters, $ \Omega\subset \mathbb{R}^{3} $ is a bounded domain with smooth boundary $ \partial\Omega $. Combining Nehari manifold method with the quantitative deformation lemma, we prove that there exists $ \mu^{\ast} > 0 $ such that above problem has at least a least energy sign-changing (or nodal) solution if $ \lambda < a\lambda_{1} $ and $ \mu > \mu^{\ast} $, where $ \lambda_{1} > 0 $ is the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. It is noticed that the nonlinearity $ \lambda u+\mu|u|^{2}u $ fails to satisfy super-linear near zero and super-three-linear near infinity, respectively.

We study the following fractional logarithmic Schrödinger equation: (−Δ)su+V(x)u=ulogu2,x∈RN,{\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N≥1N\ge 1, (−Δ)s{\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0

This paper focuses on a class of supercritical, quasi-linear Schrödinger equations. Based on the methods of invariant sets, some results about the existence and multiplicity of sign-changing solutions for supercritical equations are obtained. [ABSTRACT FROM AUTHOR]

This paper studies the nonlocal fractional analog of the famous paper of Brezis and Nirenberg [Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477]. Namely, we focus on the following model: P−Δsu−λu=α|u|p−2u+β|u|2s∗−2uinΩ,u=0inRN∖Ω,$$\begin{align*}\hskip5pc {\left(\mathcal{P}\right)} {\left\{ \def\eqcellsep{&}\begin{array}{l} {\left(-\Delta \right)}^s u-\lambda u = \alpha |u|^{p-2}u + \beta |u|^{2^*_s-2}u \quad \mbox{in}\quad \Omega ,\\ u=0\quad \mbox{in}\quad \mathbb {R}^N\setminus \Omega , \end{array} \right.}\hskip-5pc \end{align*}$$where (−Δ)s$(-\Delta)^s$ is the fractional Laplace operator, s∈(0,1)$s \in (0,1)$, with N>2s$N > 2s$, 20,λ,α∈R$\beta >0,\, \lambda , \alpha \in \mathbb {R}$, and establish the existence of nontrivial solutions and sign‐changing solutions for the problem (P)$(\mathcal{P})$. [ABSTRACT FROM AUTHOR]

In this paper, the following Schrödinger–Bopp–Podolsky system is studied - Δ u + V (x) u + ϕ u = f (u) , in R 3 , - Δ ϕ + a 2 Δ 2 ϕ = 4 π u 2 , in R 3 , where a ⩾ 0 , under general assumptions of V and f, by using an abstract critical theorem, which derived from the minimax method and the method of invariant sets of descending flow, the existence of sign-changing solution for this system is obtained, in particular, a sequence of high energy sign-changing solutions is obtained if f is odd. Moreover, we also study the asymptotic behavior of the solution as a → 0 , specifically, the sign-changing solution we find tends to the sign-changing solution of the classical Schrödinger–Poisson system as a → 0 . [ABSTRACT FROM AUTHOR]

We consider the following doubly nonlocal nonlinear logistic problem driven by the fractional p-Laplacian (- Δ) p s u = f (x , u) - ∫ Ω | u (y) | r | x - y | α d y | u (x) | r - 2 u (x) in Ω , u = 0 in R N \ Ω. Here Ω ⊂ R N (N ≥ 2) is a bounded domain with C 1 , 1 boundary ∂ Ω , s ∈ (0 , 1) , p ∈ (1 , ∞) are such that p s < N . Also p s , α # ≤ r < ∞ , where p s , α # = (2 N - α) / 2 N . Under suitable and general assumptions on the nonlinearity f, we study the existence, nonexistence, uniqueness, and regularity of weak solutions. As for applications, we treat cases of subdiffusive type logistic Choquard problem. We also consider in the superdiffusive case the Brezis-Nirenberg type problem with logistic Choquard and show the existence of a nontrivial solution for a suitable choice of λ . Finally for a particular choice of f viz. f (x , t) = λ t q - 1 with 1 < p < 2 r < q , we show the existence of at least one energy nodal solution. [ABSTRACT FROM AUTHOR]

We consider the following Schrödinger system $ \begin{equation*} \left\{ \begin{aligned} &-\Delta u_j = \sum\limits_{i = 1}^k \beta_{ij}|u_i|^3|u_j|u_j+\lambda_j|u_j|^{q-2}u_j, \ \ \ \text{in}\, \, \Omega, \\ &u_j = 0\quad\text{on}\, \, \partial\Omega, \, \, j = 1, \cdots, k \end{aligned} \right. \end{equation*} $ where $ \Omega\subset\mathbb{R}^3 $ is a bounded domain with smooth boundary. Assume $ 5 < q < 6, \, \lambda_j > 0, \, \beta_{jj} > 0, \, j = 1, \cdots, k $, $ \beta_{ij} = \beta_{ji}, \, i\neq j, i, j = 1, \cdots, k $. Note that the nonlinear coupling terms are of critical Sobolev growth in dimension $ 3 $. We prove that under an additional condition on the coupling matrix the problem has infinitely many sign-changing solutions. The result is obtained by combining the method of invariant sets of descending flow with the approach of using approximation of systems of subcritical growth.

On a smooth, closed Riemannian manifold (M , g) of dimension n ≥ 3 , we consider the stationary Schrödinger equation Δ g u + h 0 u = | u | 2 ⁎ − 2 u , where Δ g : = − div g ∇ , h 0 ∈ C 1 (M) and 2 ⁎ : = 2 n n − 2. We prove that, up to perturbations of the potential function h 0 in C 1 (M) , the sets of sign-changing solutions that are bounded in H 1 (M) are precompact in the C 2 topology. We obtain this result under the assumptions that (M , g) is locally conformally flat, n ≥ 7 and h 0 ≠ n − 2 4 (n − 1) Scal g at all points in M , where Scal g is the scalar curvature of the manifold. We then provide counterexamples in every dimension n ≥ 3 showing the optimality of these assumptions. [ABSTRACT FROM AUTHOR]