Nonlinear Schrödinger equations involving exponential critical growth with unbounded or vanishing potentials

In this paper we deal with the following class of nonlinear Schrödinger equations − Δ u + V ( | x | ) u = λ Q ( | x | ) f ( u ) , x ∈ R 2 , where λ > 0 is a real parameter, the potential V and the weight Q are radial, which can be singular at the origin, unbounded or decaying at infinity and the nonlinearity f ( s ) behaves like e α s 2 at infinity. By performing a variational approach based on a weighted Trudinger–Moser type inequality proved here, we obtain some existence and multiplicity results.

Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

Radial quasilinear elliptic problems with singular or vanishing potentials

<p style='text-indent:20px;'>In this paper we continue the work that we began in [<xref ref-type="bibr" rid="b6">6</xref>]. Given <inline-formula><tex-math id="M1">\begin{document}$ 1&lt;p&lt;N $\end{document}</tex-math></inline-formula>, two measurable functions <inline-formula><tex-math id="M2">\begin{document}$ V\left(r \right)\geq 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ K\left(r\right)&gt; 0 $\end{document}</tex-math></inline-formula>, and a continuous function <inline-formula><tex-math id="M4">\begin{document}$ A(r) &gt;0 $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M5">\begin{document}$ r&gt;0 $\end{document}</tex-math></inline-formula>), we consider the quasilinear elliptic equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u = K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where all the potentials <inline-formula><tex-math id="M6">\begin{document}$ A,V,K $\end{document}</tex-math></inline-formula> may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space <inline-formula><tex-math id="M7">\begin{document}$ X $\end{document}</tex-math></inline-formula> into the sum of Lebesgue spaces <inline-formula><tex-math id="M8">\begin{document}$ L_{K}^{q_{1}}+L_{K}^{q_{2}} $\end{document}</tex-math></inline-formula>. The nonlinearity has a double-power super <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>-linear behavior, as <inline-formula><tex-math id="M10">\begin{document}$ f(t) = \min \left\{ t^{q_1 -1}, t^{q_2 -1} \right\} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ q_1,q_2&gt;p $\end{document}</tex-math></inline-formula> (recovering the power case if <inline-formula><tex-math id="M12">\begin{document}$ q_1 = q_2 $\end{document}</tex-math></inline-formula>). With respect to [<xref ref-type="bibr" rid="b6">6</xref>], in the present paper we assume some more hypotheses on <inline-formula><tex-math id="M13">\begin{document}$ V $\end{document}</tex-math></inline-formula>, and we are able to enlarge the set of values <inline-formula><tex-math id="M14">\begin{document}$ q_1 , q_2 $\end{document}</tex-math></inline-formula> for which we get existence results.</p>

Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials

In this paper, we study the fractional Schrödinger-Poisson system ( − Δ ) s u + V ( x ) u + K ( x ) ϕ | u | q − 2 u = h ( x ) f ( u ) + | u | 2 s ∗ − 2 u , in R 3 , ( − Δ ) t ϕ = K ( x ) | u | q , in R 3 , $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} (-{\it\Delta})^{s}u+V(x)u+ K(x) \phi|u|^{q-2}u=h(x)f(u)+|u|^{2^{\ast}_{s}-2}u,&\mbox{in}~ {\mathbb R^{3}},\\ (-{\it\Delta})^{t}\phi=K(x)|u|^{q},&\mbox{in}~ {\mathbb R^{3}}, \end{array}\right. \end{array}$$ where s, t ∈ (0, 1), 3 < 4s < 3 + 2t, q ∈ (1, 2 s ∗ $\begin{array}{} \displaystyle 2^*_s \end{array}$ /2) are real numbers, (−Δ) s stands for the fractional Laplacian operator, 2 s ∗ := 6 3 − 2 s $\begin{array}{} \displaystyle 2^{*}_{s}:=\frac{6}{3-2s} \end{array}$ is the fractional critical Sobolev exponent, K, V and h are non-negative potentials and V, h may be vanish at infinity. f is a C 1-function satisfying suitable growth assumptions. We show that the above fractional Schrödinger-Poisson system has a positive and a sign-changing least energy solution via variational methods.

Hamiltonian systems of Schrödinger equations with vanishing potentials

In this paper, by means of minimax techniques involving Cerami sequences, we prove the existence of at least one pair of positive solutions for a Hamiltonian system of Schrödinger equations in [Formula: see text] with potentials vanishing at infinity and subcritical nonlinearities which are superlinear at the origin and at infinity. We establish new estimates to prove the boundedness of a Cerami sequence.