Positive solutions of a Kirchhoff-type nonlinear elliptic equation with non-local integral term on bounded domain in ℝ N , ⩾ 1, are studied by using bifurcation theory. The parameter regions existence, non-existence and uniqueness positive characterized the eigenvalues linear eigenvalue problem problem. Local global diagrams for various obtained.

In this paper, a class of generalized quasilinear Schrödinger-Maxwell systems is considered. Via the mountain pass theorem, we conclude existence positive ground state solutions when potential may vanish at infinity and nonlinear term has quasicritical growth. During process, use Coulomb energy studied by Ruiz [Arch. Ration. Mech. Anal. 198(1), 349–368 (2010)] establish convergency theorem to overcome lack compactness caused which infinity.

A sufficient condition for blowup of solutions to a class pseudo‐parabolic equations with nonlocal term is established in this paper. In virtue the potential wells method, we first extend results obtained by Xu and Su [J. Funct. Anal., 264 (12): 2732‐2763, 2013] case describe successfully behavior using energy functional, Nehari ground state stationary equation. Sequently, study boundedness convergency any global solution. Finally, achieve criterion guarantee without limit initial...

In this paper, we consider the existence of solutions to a quadratically coupled Schrödinger systems −Δu=μ1u+2αu v in RN,−Δv=μ2v+αu2 RN under condition ∫RNu2=a,∫RNv2=b. Here N=1, 2, α>0 and a, b>0 are fixed. systems, μ1 μ2 unknown. We prove that there exists solution (μ1,μ2,u~,v~) with μ1<0, μ2<0, u~,v~∈C2(RN) being positive, radially symmetric, decreasing r=|x|. Our argument is heavily based on rearrangement techniques.

In this paper, we give some properties about the $(2,p)$ -Laplacian operator ( $p>1$ , $p\ne2$ ), and consider existence of solutions to two kinds partial differential equations related by those properties. Specifically, establish an result positive using fixed point index theory nodal via quantitative deformation lemma.

In this paper, we study the following quadratically coupled Schrödinger system: $\begin{equation*}\left\{\begin{array}{ll}-\Delta u+\lambda_1u=\mu_1u^2+2\alpha uv+\gamma v^2, &amp; \mbox{in }\Omega,\\-\Delta v+\lambda_2v=\mu_2v^2+2\gamma uv+\alpha u^2, }\Omega,\\u=v=0, \mbox{on }\partial\Omega,\end{array}\right.\end{equation*}$ where $\Omega\subset\mathbb{R}^6$ is a smooth bounded domain, $-\lambda (\Omega) &lt; \lambda_1, \lambda_2 0, \mu_1, \mu_2, \alpha, \gamma&gt;0$, and $\lambda...

In this work, we investigate the (2, p)-Laplacian equation −Δu − Δpu = f(x, u) in Ω with boundary condition u 0 on ∂Ω, where is a smooth bounded domain RN, p &amp;gt; 2, and nonlinearity f has extension property at both zero infinity points. We observe that above admits least two positive solutions, owing to mountain pass theorem Ekeland’s variational principle.