This paper is dedicated to studying the existence of normalized solutions for a class of Chern-Simons-Schrödinger system, where the nonlinearity possesses critical exponential growth of ...Trudinger-Moser type. Under some weak assumptions, we obtain several new existence results by employing more delicate estimates and analytical technical. Our results improve and complement the works of Yao et al. (2023) 25 and Yuan et al. (2022) 27.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
This paper is concerned with the following planar Schrödinger-Poisson system{−Δu+V(x)u+ϕu=f(x,u),x∈R2,Δϕ=u2,x∈R2, where V∈C(R2,0,∞)) is axially symmetric and f∈C(R2×R,R) is of subcritical or critical ...exponential growth in the sense of Trudinger-Moser. We obtain the existence of a nontrivial solution or a ground state solution of Nehari-type and infinitely many solutions to the above system under weak assumptions on V and f. Our theorems extend the results of Cingolani and Weth Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016) 169-197 and of Du and Weth Nonlinearity, 30 (2017) 3492-3515 and Chen and Tang J. Differential Equations, 268 (2020) 945-976, where f(x,u) has polynomial growth on u. In particular, some new tricks and approaches are introduced to overcome the double difficulties resulting from the appearance of both the convolution ϕ2,u(x) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity f(x,u).
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form(0.1)−Δu+V(|x|)u=Q(|x|)f(u)+λg(|x|,u),x∈R2, where the nonlinear term f(s) has critical exponential ...growth which behaves like eαs2, g(r,s) is a concave term on s, the radial potentials V,Q:R+→R are unbounded, singular at the origin or decaying to zero at infinity and λ>0 is a parameter. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality in the working space of the associated with the energy functional related to the above problem. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weak assumptions. Our results show that the presence of the concave term (i.e. λ>0) is very helpful to the existence of nontrivial solutions for Eq. (0.1) in one sense.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In the present paper, we study the following planar Choquard equation:{−Δu+V(x)u=(Iα⁎F(u))f(u),x∈R2,u∈H1(R2), where V(x) is an 1-periodic function, Iα:R2→R is the Riesz potential and f(t) behaves ...like ±eβt2 as t→±∞. A direct approach is developed in this paper to deal with the problems with both critical exponential growth and strongly indefinite features when 0 lies in a gap of the spectrum of the operator −△+V. In particular, we find nontrivial solutions for the above equation with critical exponential growth, and establish the existence of ground states and geometrically distinct solutions for the equation when the nonlinearity has subcritical exponential growth. Our results complement and generalize the known ones in the literature concerning the positive potential V to the general sign-changing case, such as, the results of de Figueiredo-Miyagaki-Ruf (1995) 16, of Alves-Cassani-Tarsi-Yang (2016) 4, of Ackermann (2004) 1, and of Alves-Germano (2018) 5.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
This paper focuses on the study of ground states and nontrivial solutions for the following Hamiltonian elliptic system:{−Δu+V(x)u=f1(x,v),x∈R2,−Δv+V(x)v=f2(x,u),x∈R2, where V∈C(R2,(0,∞)) and ...f1,f2:R2×R→R have critical exponential growth. The strongly indefinite features together with the critical exponent bring some new difficulties in our analysis. In this paper, we develop a direct approach and use an approaching argument to seek Cerami sequences for the energy functional and estimate the minimax levels of such sequences. In particular, under some general assumptions imposed on the nonlinearity fi, we obtain the existence of ground states and nontrivial solutions for the above system as well as the following system in bounded domain,{−Δu=f1(v),x∈Ω,−Δv=f2(u),x∈Ω,u=0,v=0,x∈∂Ω. Our results improve and extend the related results of de Figueiredo-do Ó-Ruf (2004) 20; (2011) 21, of Lam-Lu (2014) 30, and of de Figueiredo-do Ó-Zhang (2020) 22.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
This paper is concerned with the following planar Schrödinger-Poisson system with zero mass potential{−Δu+ϕu=Q(|x|)f(u),x∈R2,Δϕ=u2,x∈R2, where Q∈C((0,∞),(0,∞)) may be singular or unbound and f∈C(R,R) ...is of critical exponential growth and there is no monotonicity restriction on f(u)/u3. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality within the working space associated with the energy functional relevant to the aforementioned problem, which is different from Albuquerque et al. (2021) 5. By combining the variational methods and delicate estimates, we prove the existence of a non-trivial mountain-pass solution to the above system under mild assumptions on Q and f.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
This paper is concerned with the following planar Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(u),x∈R2,Δϕ=u2,x∈R2,where V∈C(R2,0,∞)) is axially symmetric and f∈C(R,R) has critical exponential growth in ...the sense of Trudinger–Moser. This system can be converted into the integro-differential equation with logarithmic convolution potential. We prove the existence of axially symmetric solutions by some new useful estimates on logarithmic convolution potential. Our result not only improves the ones of Chen–Tang (2020), but covers the zero mass case that V(x)≡0.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In the present paper, we are concerned with the following fractional p-Laplacian Choquard logarithmic equation(−Δ)psu+V(x)|u|p−2u+(ln|⋅|⁎|u|p)|u|p−2u=(∫RNF(y,u)|y|β|x−y|μdy)f(x,u)|x|βinRN, where ...N=sp≥2, s∈(0,1), 0<μ<N, β≥0, 2β+μ≤N and (−Δ)ps denotes the fractional p-Laplace operator, the potential V∈C(RN,0,∞)), and f:RN×R→R is continuous. Under mild conditions and combining variational and topological methods, we obtain the existence of axially symmetric solutions both in the exponential subcritical case and in the exponential critical case. We point out that we take advantage of some refined analysis techniques to get over the difficulty carried by the competition of the Choquard logarithmic term and the Stein-Weiss nonlinearity. Moreover, in the exponential critical case, we extend the nonlinearities to more general cases compared with the existing results.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
In this paper, we develop new proof techniques and analytical methods to prove the existence of ground state solutions for the following planar Schrödinger-Poisson system with zero ...mass{−Δu+ϕu=f(u),x∈R2,Δϕ=2πu2,x∈R2, where f∈C(R,R) has the critical exponential growth at infinity and there is no monotonicity restriction on f(u)/u3. In particular, by using delicate estimates we obtain a desired upper bound for the Mountain Pass level just with the optimal asymptotic condition κ=liminf|t|→∞t2F(t)eα0t2>0 to restore the compactness in the presence of critical exponential growth, which significantly improves analogous assumptions on asymptotic behavior of t2F(t)eα0t2 or tf(t)eα0t2 at infinity in the previous works. Moreover, we use a different approach from the one of Du and Weth (2017) 21 dealing with the power nonlinearities to establish the Pohozaev type identity, which not only allows critical exponential growth nonlinearities, but also deals with the non-autonomous case containing a linear term V(x)u in the first equation, both of which are not covered in the existing literature. To our knowledge, there has not been any work in the literature on the subject, even for the simpler equation: −Δu=f(u) in R2.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
