Multi-bump Type Nodal Solutions for a Fractional p-Laplacian Logarithmic Schrödinger Equation with Deepening Potential Well

Multi-bump type nodal solutions having a prescribed number of nodal domains: II

In this paper, we are concerned with the existence and multiplicity of multi-bump type nodal solutions for the following logarithmic Schrodinger equation $$ \left\{ \begin{array}{ll} -\Delta u+ \lambda V(x)u=u \log u^2, &{}\quad \text{ in } \quad {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), \\ \end{array} \right. $$ where $$N \ge 1$$ , $$\lambda >0$$ is a real parameter and the nonnegative continuous function $$V: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$$ has a potential well $$\Omega : =\text {int}\, V^{-1}(0)$$ which possesses k disjoint bounded components $$\Omega =\bigcup _{j=1}^{k}\Omega _{j}$$ . Using the variational methods, we prove that if the parameter $$\lambda >0$$ is large enough, then the equation has at least $$2^{k}-1$$ multi-bump type nodal solutions.

Multi-bump type nodal solutions having a prescribed number of nodal domains: I

In this paper, we establish existence and multiplicity of multi-bump type nodal solutions for the following class of problems $$ -\Delta u + (\lambda V(x)+ 1)u= f(u), \quad u> 0 \quad \text{in } {\mathbb R}^N, $$ where $N \geq 1$, $\lambda \in (0, \infty), f$ is a continuous function with subcritical growth and $V\colon {\mathbb R}^N \rightarrow {\mathbb R} $ is a continuous function verifying some hypotheses.

We construct multi-bump nodal solutions of the elliptic equationin $H^1_0(\varOmega)$, when μ is large, under appropriate assumptions, for f superlinear and subcritical and such that the eigenvalues of the associated linearized operator on $H^1_0(\{x\in\varOmega:a(x)>0\})$ at zero, u ↦ u − λ(−Δ)−1(a+u), are positive. The solutions are of least energy in some Nehari-type set defined by imposing suitable conditions on orthogonal components of functions in $H^1_0(\varOmega)$.

In this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problemswhereλ ∈(0, ∞),fis a continuous function with exponential critical growth andV: ℝ2→ℝ is a continuous function verifying some hypotheses.

Extremal constant sign solutions and nodal solutions for the fractional p-Laplacian

In this paper the results of some researches concerning Scalar Field Equations are summarized. The interest is focused on the question of existence and multiplicity of stationary solutions, so the model equation \( -\Delta u + a(x)u = |u|^{p-1}u \; \; \rm{in} \; \mathbb{R}^{N} \) is considered. The difficulties and the ideas introduced to face them as well as some well known results are discussed. Some recent advances concerning existence and multiplicity of multi-bump solutions are described in more detail.KeywordsElliptic equations in RNvariational methodsmulti-bump solutionsinfinitely many positive and nodal solutions.

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...