We investigate a
class of nonlinear biharmonic equations with
p-Laplacian
class="formula" id="fm0010">
where
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si2.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=f046146b18f79cb805721277d205f9cf" title="Click to view the MathML source">N≥1class="mathContainer hidden">class="mathCode">,
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si175.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=ae327ddfa9a672086c0bfffbf06313a6" title="Click to view the MathML source">β∈Rclass="mathContainer hidden">class="mathCode">,
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si242.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=6002b7d3b70cc9051c56c014ed38311f" title="Click to view the MathML source">λ>0class="mathContainer hidden">class="mathCode"> is a parameter and
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si11.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=3fbb2fcca900127c244cf352886b0175" title="Click to view the MathML source">Δpu=div(|∇u|p−2∇u)class="mathContainer hidden">class="mathCode"> with
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si6.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=70fd0c5438e947cee324b071a40d14e0" title="Click to view the MathML source">p≥2class="mathContainer hidden">class="mathCode">. Unlike most other papers on this problem, we replace Laplacian with
p-Laplacian and allow
β to be negative. Under some suitable assumptions on
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si7.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=7f50bc09cb42ee5a739c1aef0a8171d8" title="Click to view the MathML source">V(x)class="mathContainer hidden">class="mathCode"> and
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616303266&_mathId=si8.gif&_user=111111111&_pii=S0022039616303266&_rdoc=1&_issn=00220396&md5=ee13f4fa526ec2718cb949ecd8aa6b0a" title="Click to view the MathML source">f(x,u)class="mathContainer hidden">class="mathCode">, we obtain the existence and multiplicity of nontrivial solutions for
λ large enough. The proof is based on variational methods as well as Gagliardo–Nirenberg inequality.